#504405

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

622 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)(7,10)(8,9)(11,12)(13,14,17,19)(15,20,24,34)(16,26,28,21)(18,22,30,33)(23,39,40,35)(25,31,37,43)(27,41,44,36)(29,42,38,32);(2,4)(13,15,17,24)(14,20,19,34)(16,27,28,44)(18,31,30,43)(21,36,26,41)(22,37,33,25)(23,38,40,29)(32,35,42,39);(6,9)(10,12)(13,16,17,28)(14,21,19,26)(15,23,24,40)(18,32,30,42)(20,35,34,39)(22,38,33,29)(25,41,37,36)(27,43,44,31);(2,4)(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 411601);