#6389173

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1990 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^d=ei

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, 6296369);