#7608217

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1283 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)(17,18,24,30)(19,28,33,43)(20,34,38,26)(21,29,23,27)(22,40,32,25)(31,46,35,48)(36,45,39,47)(37,44,41,42);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,22,24,33,23,32)(18,25,27,43,30,40,29,28)(20,35,39,37,38,31,36,41)(26,44,47,48,34,42,45,46);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,26)(19,31)(21,39)(22,41)(23,36)(24,38)(25,42)(27,47)(28,48)(29,45)(30,34)(32,37)(33,35)(40,44)(43,46);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(17,21,24,23)(18,27,30,29)(19,22,33,32)(20,36,38,39)(25,43,40,28)(26,45,34,47)(31,37,35,41)(42,48,44,46);(9,13,14,12)(10,11,15,16)(17,22,21,33,24,32,23,19)(18,28,27,25,30,43,29,40)(20,37,39,31,38,41,36,35)(26,46,47,44,34,48,45,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=ghi,d^2=g,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,e^c=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, 7515413);