#501546

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

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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:

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

Polycyclic presentation:

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

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, 408742);