#507273

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=efgi,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^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414469);