#436466

order	512
nilpotency class	3
isoclinism	isoclinic class of #436465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15561
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1143 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13,17,38)(4,9)(5,22,25,10)(6,11)(7,12)(8,16,31,26)(14,42,18,41)(15,45)(19,49,52,33)(20,34)(21,35)(23,30)(24,53,56,37)(27,39)(28,40)(29,44,32,47)(36,46)(43,60)(48,63)(50,59)(51,62,55,61)(54,58)(57,64);(1,3,6,26,7,17,28,16)(2,8,11,38,12,31,40,13)(4,18,20,44,21,14,27,47)(5,23,24,15,25,54,56,46)(9,32,34,42,35,29,39,41)(10,36,37,30,22,45,53,58)(19,50,51,48,52,64,55,43)(33,60,61,59,49,63,62,57);(1,4,7,21)(2,9,12,35)(3,14,17,18)(5,19,25,52)(6,27,28,20)(8,29,31,32)(10,33,22,49)(11,39,40,34)(13,41,38,42)(15,43,46,48)(16,47,26,44)(23,50,54,64)(24,55,56,51)(30,57,58,59)(36,60,45,63)(37,62,53,61);(1,5)(2,10)(3,15,17,46)(4,19)(6,24)(7,25)(8,30,31,58)(9,33)(11,37)(12,22)(13,36,38,45)(14,43,18,48)(16,23,26,54)(20,51)(21,52)(27,55)(28,56)(29,57,32,59)(34,61)(35,49)(39,62)(40,53)(41,60,42,63)(44,64,47,50)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286126, #72286206, #72286235
maximal normal subgroup of	#72286126, #72286206, #72286235

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343662);