#415917

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323113);