#7620935

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1290 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)(11,14)(12,13)(15,16)(17,18)(19,30)(20,26)(21,27)(22,37)(23,29)(24,28)(25,33)(31,45)(32,40)(34,41)(35,44)(36,43)(38,42)(39,46)(47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,30)(20,31,36,46)(21,22,32,24)(26,39,43,45)(27,28,40,37)(34,38,47,35)(41,44,48,42);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,34)(22,38)(23,36)(24,35)(25,39)(27,41)(28,44)(29,43)(30,45)(32,47)(33,46)(37,42)(40,48);(10,13)(14,16)(17,21,24,19,23,32,22,33)(18,27,28,30,29,40,37,25)(20,34,35,31,36,47,38,46)(26,41,44,45,43,48,42,39);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,37)(19,32,33,21)(20,35,36,38)(25,40,30,27)(26,42,43,44)(31,34,46,47)(39,41,45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=gh,e^2=fg,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,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, 7528131);