#475910

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383106);