#7595323

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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,34)(21,36)(22,27)(23,29)(24,28)(25,32)(26,39)(31,43)(33,42)(35,46)(37,41)(38,45)(40,44)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,25,27,30)(20,23,33,24)(21,37,38,35)(26,43,44,48)(28,42,29,34)(31,39,47,40)(36,46,45,41);(10,14)(11,16)(17,20,23,32,22,33,24,19)(18,25,28,34,27,30,29,42)(21,35,39,47,38,37,40,31)(26,41,36,48,44,46,45,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,39,22,38,24,40)(18,26,28,36,27,44,29,45)(19,31,20,35,32,47,33,37)(25,41,34,48,30,46,42,43);(2,5)(6,8)(10,14)(13,15)(18,27)(21,38)(25,30)(28,29)(31,47)(34,42)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fgi,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, 7502519);