#507253

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,35)(29,36)(31,43)(32,44)(33,39)(34,41)(37,45)(38,48)(40,46)(42,47);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,28,45,33,48,34,29,20,30,21,31,22,32,46,47)(18,23,35,43,39,44,41,42,24,27,25,37,26,38,40,36);(10,13)(14,15)(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(27,42)(28,46)(30,47)(31,32)(35,41)(37,44)(38,43)(45,48);(2,6)(5,7)(9,11,12,16)(10,14,13,15)

Polycyclic presentation:

2^9|b^2=efgi,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 414449);