#415028

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322224);