#494660

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1258 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)(5,6,9,15,12,17,11,35)(7,18,20,48,22,45,28,49)(8,23,26,44,27,42,24,14)(10,30,29,37,34,61,33,16)(13,36,39,21,41,55,40,31)(19,50,53,63,54,67,51,47)(25,43,60,58,32,46,59,66)(38,52,64,68,65,56,57,62);(5,7,10,31,11,20,33,36,12,22,34,21,9,28,29,55)(6,13,37,18,15,40,61,49,17,41,16,45,35,39,30,48)(8,19,32,52,26,51,59,62,27,54,25,56,24,53,60,68)(14,38,43,47,42,64,66,50,44,65,46,63,23,57,58,67);(2,4)(5,8)(6,14)(7,19)(9,24)(10,32)(11,26)(12,27)(13,38)(15,42)(16,46)(17,44)(18,47)(20,51)(21,56)(22,54)(23,35)(25,34)(28,53)(29,60)(30,58)(31,52)(33,59)(36,62)(37,43)(39,57)(40,64)(41,65)(45,63)(48,67)(49,50)(55,68)(61,66);(5,9,12,11)(6,15,17,35)(7,20,22,28)(8,24,27,26)(10,29,34,33)(13,39,41,40)(14,42,44,23)(16,30,37,61)(18,48,45,49)(19,51,54,53)(21,55,31,36)(25,59,32,60)(38,57,65,64)(43,66,46,58)(47,67,63,50)(52,62,56,68)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401856);