#415874

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

1622 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)(15,20,26,35)(16,29,31,40)(17,23,18,22)(21,33,39,32)(25,41,42,38)(27,37,28,36)(30,44,43,34);(6,9)(10,11)(13,15)(14,20)(16,30,31,43)(17,27)(18,28)(19,26)(21,38,39,41)(22,36)(23,37)(24,35)(25,33,42,32)(29,34,40,44);(2,4)(13,16)(14,21)(15,25)(17,33)(18,32)(19,31)(20,34)(22,40)(23,29)(24,39)(26,42)(27,30)(28,43)(35,44)(36,38)(37,41);(5,7,8,12)(6,10,9,11)(15,26)(20,35)(25,42)(27,28)(30,43)(34,44)(36,37)(38,41)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,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, 323070);