#380595

order	512
nilpotency class	3
isoclinism	isoclinic class of #380594
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14271
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1428 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286082, #72286206, #72286253
maximal normal subgroup of	#72286082, #72286206, #72286253

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287791);