#6389233

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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

1995 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)(15,24)(16,21)(17,22)(18,33)(19,23)(20,26)(25,41)(27,35)(28,44)(29,38)(30,40)(31,36)(32,39)(34,43)(37,42);(1,3,4,2)(5,7)(6,9)(8,12)(10,11);(13,15,18,26,19,27,17,31)(14,20,22,24,23,36,33,35)(16,28,29,42,30,25,32,43)(21,37,39,44,40,34,38,41);(7,12)(9,11)(13,16,19,30)(14,21,23,40)(15,25,27,28)(17,32,18,29)(20,34,36,37)(22,39,33,38)(24,41,35,44)(26,43,31,42);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,36,24)(21,38,40,39)(25,42,28,43)(34,41,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296429);