#7617002

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 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)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,32)(21,36)(22,27)(23,26)(24,31)(25,35)(29,39)(30,40)(33,42)(34,43)(37,44)(38,41)(45,48)(46,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,38)(25,42)(26,40)(27,28)(29,35)(32,39)(34,46)(36,41)(37,45)(43,48)(44,47);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,39)(26,44)(27,32)(28,42)(30,46)(31,33)(36,43)(38,45)(40,48)(41,47);(10,14)(13,15)(17,19,21,38,22,31,23,30)(18,24,36,40,27,28,26,41)(20,33,37,46,35,29,34,45)(25,42,43,47,32,39,44,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,36)(19,30,31,38)(20,34,35,37)(24,40,28,41)(25,43,32,44)(29,45,33,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7524198);