#502893

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

891 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)(15,23)(16,19)(17,30)(18,31)(20,24)(21,25)(22,29)(26,34)(27,43)(28,36)(32,39)(33,42)(35,44)(37,41)(38,40);(6,9)(10,11)(14,19)(15,24)(17,25)(18,29)(20,34)(21,36)(23,32)(26,39)(27,44)(28,42)(30,33)(35,38)(37,43)(40,41);(1,2,3,4)(13,15,25,41,18,27,28,39,16,26,42,44,29,40,17,24)(14,20,30,38,22,35,33,34,19,32,36,43,31,37,21,23);(5,7,8,12)(6,10,9,11)(14,19)(20,32)(21,33)(22,31)(23,34)(30,36)(35,37)(38,43)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410089);