#7595319

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(19,28)(20,31)(21,34)(22,26)(23,38)(24,30)(25,36)(27,37)(29,44)(32,43)(33,42)(35,48)(39,47)(40,46)(41,45);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,26,28)(20,23,33,37)(21,32,35,47)(25,40,41,44)(27,42,38,31)(29,45,46,36)(34,39,48,43);(10,14)(11,16)(17,20,23,30,22,33,37,19)(18,24,27,31,26,28,38,42)(21,32,36,46,35,47,45,29)(25,39,34,40,41,43,48,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,36,22,35,37,45)(18,25,27,34,26,41,38,48)(19,29,20,32,30,46,33,47)(24,39,31,40,28,43,42,44);(2,5)(6,8)(10,14)(13,15)(18,26)(21,35)(24,28)(27,38)(29,46)(31,42)(32,47)(36,45)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 7502515);