#504602

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

486 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ei,d^2=f,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, 411798);