#7600618

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(17,18,23,28)(19,26,31,42)(20,25,32,39)(21,35,30,24)(22,37,36,27)(29,40,43,48)(33,46,44,38)(34,47,45,41);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,30,23,31,36,21)(18,24,27,26,28,35,37,42)(20,29,34,44,32,43,45,33)(25,38,41,40,39,46,47,48);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,38)(26,40)(27,41)(28,39)(30,44)(31,43)(35,46)(36,45)(37,47)(42,48);(2,6)(3,8)(10,14)(13,16)(20,32)(25,39)(29,43)(33,44)(34,45)(38,46)(40,48)(41,47);(2,6)(5,7)(17,21,36,31,23,30,22,19)(18,26,37,24,28,42,27,35)(20,33,45,43,32,44,34,29)(25,40,47,38,39,48,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,e^2=fgi,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, 7507814);