#7595834

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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:

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

Polycyclic presentation:

2^9|c^2=fhi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7503030);