#507159

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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

490 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414355);