#504224

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,15)(11,14,12,16)(17,18,20,24)(19,27,29,23)(21,25,32,39)(22,26,33,40)(28,36,46,41)(30,45,47,37)(31,44,35,38)(34,43,48,42);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,29)(21,28)(22,35)(24,27)(25,41)(26,44)(30,34)(31,33)(32,46)(36,39)(37,43)(38,40)(42,45)(47,48);(9,12,13,11)(10,14,15,16)(18,24)(19,28)(21,34)(22,33)(23,36)(25,42)(27,41)(29,46)(30,31)(32,48)(35,47)(37,44)(38,45)(39,43);(1,3,4,8)(2,6,5,7)(18,24)(23,27)(25,39)(26,40)(36,41)(37,45)(38,44)(42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=ei,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, 411420);