#7614788

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(19,25)(20,32)(21,35)(22,27)(23,28)(24,29)(26,38)(30,46)(31,41)(33,40)(34,45)(36,48)(37,44)(39,43)(42,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,23)(21,36)(22,31)(24,34)(26,42)(27,41)(28,32)(29,45)(30,39)(33,37)(35,48)(38,47)(40,44)(43,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,19)(18,25,28,32)(21,33,38,30)(22,31,24,34)(26,40,35,46)(27,45,29,41)(36,39,47,37)(42,43,48,44);(10,14)(13,15)(17,21,24,39,23,38,22,37)(18,26,29,44,28,35,27,43)(19,30,31,36,20,33,34,47)(25,40,41,42,32,46,45,48);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,31,20,34)(21,37,38,39)(25,41,32,45)(26,43,35,44)(30,36,33,47)(40,42,46,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=h,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,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, 7521984);