#7608030

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1285 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=g,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,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515226);