#7609725

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,34)(20,24,31,37)(21,25,32,38)(27,33,40,44)(29,35,41,45)(30,36,42,46)(39,43,47,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(20,30)(21,32)(24,36)(25,38)(27,39)(29,41)(31,42)(33,43)(35,45)(37,46)(40,47)(44,48);(10,14)(11,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,21,30,22,31,32,42)(18,24,25,36,26,37,38,46)(19,27,29,39,28,40,41,47)(23,33,35,43,34,44,45,48);(2,7)(6,8)(19,28)(23,34)(27,40)(29,41)(33,44)(35,45)(39,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7516921);