#7600411

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

1361 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)(4,6)(5,8)(17,18,24,30)(19,28,33,42)(20,34,38,26)(21,29,23,27)(22,40,32,25)(31,47,35,45)(36,44,39,46)(37,41,48,43);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,22,24,33,23,32)(18,25,27,42,30,40,29,28)(20,35,39,48,38,31,36,37)(26,43,46,45,34,41,44,47);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,26)(19,31)(21,39)(22,37)(23,36)(24,38)(25,41)(27,46)(28,45)(29,44)(30,34)(32,48)(33,35)(40,43)(42,47);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,27,30,29)(19,22,33,32)(20,36,38,39)(25,42,40,28)(26,44,34,46)(31,48,35,37)(41,45,43,47);(2,6)(5,7)(17,22,21,33,24,32,23,19)(18,28,27,25,30,42,29,40)(20,37,39,35,38,48,36,31)(26,45,46,41,34,47,44,43)

Polycyclic presentation:

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

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, 7507607);