#493552

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,17,12,15,11,33)(7,18,21,47,22,46,34,43)(8,23,25,39,26,41,32,14)(9,27,29,42,31,44,30,16)(13,28,37,53,36,60,38,20)(19,48,50,65,52,66,51,45)(24,40,57,55,58,64,61,59)(35,49,54,67,62,68,63,56);(5,7,12,22)(6,13,15,36)(8,19,26,52)(9,28,31,60)(10,21,11,34)(14,35,39,62)(16,43,42,47)(17,37,33,38)(18,44,46,27)(20,29,53,30)(23,54,41,63)(24,56,58,67)(25,50,32,51)(40,48,64,66)(45,55,65,59)(49,61,68,57);(2,4)(5,8)(6,14)(7,19)(9,24)(10,32)(11,25)(12,26)(13,35)(15,39)(16,40)(17,41)(18,45)(20,49)(21,51)(22,52)(23,33)(27,59)(28,56)(29,61)(30,57)(31,58)(34,50)(36,62)(37,63)(38,54)(42,64)(43,48)(44,55)(46,65)(47,66)(53,68)(60,67);(6,15)(13,36)(14,39)(16,42)(17,33)(18,46)(23,41)(27,44)(35,62)(37,38)(40,64)(43,47)(45,65)(48,66)(54,63)(55,59)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281756, #72282032, #72282220
maximal normal subgroup of	#72281756, #72282032, #72282220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400748);