#7601225

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 7508421);