#7613059

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,36)(22,28)(23,27)(24,31)(29,41)(30,39)(32,46)(33,48)(34,44)(35,43)(37,40)(38,45)(42,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,30)(22,31)(23,37)(25,28)(26,42)(27,39)(29,35)(33,45)(34,47)(36,40)(38,43)(41,48)(44,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,25)(20,29)(21,37)(22,31)(23,30)(24,28)(26,41)(27,39)(32,35)(33,47)(34,45)(36,40)(38,44)(42,48)(43,46);(10,14)(13,16)(17,20,21,33,22,34,23,35)(18,26,36,48,28,44,27,43)(19,29,37,47,31,45,30,32)(24,38,39,46,25,41,40,42);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,27,28,36)(19,30,31,37)(20,33,34,35)(24,39,25,40)(26,43,44,48)(29,32,45,47)(38,46,41,42)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=hi,e^2=f,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, 7520255);