#7595750

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,24,30)(19,25,20,34)(21,36,40,45)(22,28,23,29)(26,33,42,32)(27,39,46,38)(31,43,35,44)(37,48,47,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,20)(18,25,30,34)(21,37,40,47)(22,32,23,33)(26,29,42,28)(27,44,46,43)(31,38,35,39)(36,48,45,41);(10,14)(11,16)(17,20,23,33,24,19,22,32)(18,26,28,25,30,42,29,34)(21,31,39,37,40,35,38,47)(27,44,45,48,46,43,36,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,38,24,40,23,39)(18,27,29,36,30,46,28,45)(19,31,33,47,20,35,32,37)(25,41,26,43,34,48,42,44);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,20,33)(21,38,40,39)(25,26,34,42)(27,45,46,36)(31,37,35,47)(41,43,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=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, 7502946);