#502898

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 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:

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

Polycyclic presentation:

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

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, 410094);