#422294

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 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:

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

Polycyclic presentation:

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

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, 329490);