#427386

order	512
nilpotency class	3
isoclinism	isoclinic class of #427385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15369
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

1208 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)(3,8)(4,17,19,9)(5,21,23,51)(6,12)(7,11)(10,24,34,55)(13,27,38,42)(14,40,22,52)(15,30)(16,29)(18,45)(20,44,48,32)(25,36)(26,35)(28,41,33,54)(31,47)(37,60)(39,57,49,58)(43,62)(46,50)(53,64)(56,63)(59,61);(1,3,7,15)(2,8,11,30)(4,13,19,38)(5,22,23,14)(6,25,26,16)(9,27,17,42)(10,33,34,28)(12,36,35,29)(18,37,46,53)(20,49,48,39)(21,40,51,52)(24,41,55,54)(31,56,43,59)(32,58,44,57)(45,60,50,64)(47,63,62,61);(1,4)(2,9)(3,13)(5,18)(6,20)(7,19)(8,27)(10,31)(11,17)(12,32)(14,37)(15,38)(16,39)(21,50)(22,53)(23,46)(24,47)(25,49)(26,48)(28,56)(29,57)(30,42)(33,59)(34,43)(35,44)(36,58)(40,60)(41,61)(45,51)(52,64)(54,63)(55,62);(1,5)(2,10,11,34)(3,14,15,22)(4,18)(6,24)(7,23)(8,28)(9,31,17,43)(12,21,35,51)(13,37,38,53)(16,41,25,54)(19,46)(20,47)(26,55)(27,56)(29,40)(30,33)(32,50,44,45)(36,52)(39,61,49,63)(42,59)(48,62)(57,60)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285951, #72286082, #72286176
maximal normal subgroup of	#72285951, #72286082, #72286176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334582);