#507254

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

532 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)(19,24)(20,25)(21,27)(22,28)(23,26)(29,36)(30,38)(31,39)(32,37)(33,40)(34,41)(35,42)(43,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,29,43,35,20,33,45,23,32,21,30,22,31,44,34)(18,24,36,46,42,25,40,48,26,37,27,38,28,39,47,41);(10,13)(14,15)(17,20,23,31)(18,25,26,39)(19,22,32,35)(21,30,29,43)(24,28,37,42)(27,38,36,46)(33,34,44,45)(40,41,47,48);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(18,26)(24,37)(25,39)(27,36)(28,42)(38,46)(40,47)(41,48)

Polycyclic presentation:

2^9|b^2=efgi,c^2=i,d^2=f,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, 414450);