#437804

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1334 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:

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

Polycyclic presentation:

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

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, 345000);