#7616090

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

1371 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,39)(20,25,35,43)(21,26,30,38)(22,27,36,44)(29,37,45,47)(32,40,46,48)(33,41,34,42);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,37)(26,44)(27,38)(28,39)(32,33)(34,46)(35,45)(40,41)(42,48)(43,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,25)(19,29)(21,32)(22,34)(23,35)(24,37)(26,40)(27,42)(28,43)(30,46)(31,45)(33,36)(38,48)(39,47)(41,44);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,21,36,19,23,30,22,31)(18,26,44,24,28,38,27,39)(20,32,33,29,35,46,34,45)(25,40,41,37,43,48,42,47);(2,6)(7,8)(10,14)(13,15)(17,22,23,36)(18,27,28,44)(19,30,31,21)(20,33,35,34)(24,38,39,26)(25,41,43,42)(29,32,45,46)(37,40,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fhi,e^2=i,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,e^d=ei

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, 7523286);