#6388384

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(6,9)(7,11)(13,14,17,22,18,21,28,34)(15,19,24,31,25,30,37,41)(16,20,27,33,26,32,38,42)(23,29,36,40,35,39,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,24,18,25,28,37)(14,19,22,31,21,30,34,41)(16,23,27,36,26,35,38,43)(20,29,33,40,32,39,42,44);(2,4)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,32)(22,33)(24,36)(25,35)(28,38)(30,39)(31,40)(34,42)(37,43)(41,44);(6,9)(10,12)(14,21)(16,26)(19,30)(22,34)(23,35)(27,38)(31,41)(36,43);(5,7,8,11)(6,10,9,12)(16,26)(20,32)(23,35)(27,38)(29,39)(33,42)(36,43)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295580);