#7594526

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=h,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 7501722);