#504138

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

570 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=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, 411334);