#7615553

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,26)(21,37)(22,28)(23,29)(24,30)(27,32)(31,39)(33,40)(34,47)(35,42)(36,43)(38,44)(41,45)(46,48);(2,6)(7,8)(11,16)(13,15)(17,19,24,33)(18,25,30,40)(20,31,36,46)(21,22,32,23)(26,39,43,48)(27,29,37,28)(34,38,45,35)(41,42,47,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,34)(22,38)(23,35)(24,36)(25,39)(27,41)(28,44)(29,42)(30,43)(32,45)(33,46)(37,47)(40,48);(10,14)(13,15)(17,21,23,19,24,32,22,33)(18,27,29,40,30,37,28,25)(20,34,35,31,36,45,38,46)(26,41,42,48,43,47,44,39);(1,4,5,3)(2,7,6,8)(17,22,24,23)(18,28,30,29)(19,32,33,21)(20,35,36,38)(25,27,40,37)(26,42,43,44)(31,34,46,45)(39,47,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522749);