#455474

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
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

1165 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)(5,6,10,9)(7,12,8,11)(13,14,18,22,17,21,32,39)(15,23,26,36,25,19,27,35)(16,28,31,38,30,40,33,20)(24,37,43,34,42,44,29,41);(5,7,10,8)(6,11,9,12)(13,15,17,25)(14,19,21,23)(16,29,30,43)(18,26,32,27)(20,37,38,44)(22,35,39,36)(24,33,42,31)(28,34,40,41);(5,8,10,7)(6,11,9,12)(13,16,17,30)(14,20,21,38)(15,24,25,42)(18,33,32,31)(19,34,23,41)(22,40,39,28)(26,29,27,43)(35,37,36,44);(2,4)(5,9,10,6)(7,12,8,11)(16,30)(20,38)(24,42)(28,40)(29,43)(31,33)(34,41)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362670);