#475881

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1000 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 383077);