#7592217

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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,34)(21,39)(22,29)(23,28)(24,27)(25,33)(26,36)(31,46)(32,41)(35,45)(37,44)(38,43)(40,47)(42,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,35)(21,24)(22,32)(23,33)(26,42)(27,41)(28,30)(29,39)(31,36)(34,40)(37,47)(38,48)(43,46)(44,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,36,23,37,24,38)(18,26,27,34,28,43,29,44)(19,31,21,35,33,48,32,47)(25,40,39,42,30,45,41,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,33,23,32,24,19)(18,25,27,39,28,30,29,41)(20,35,36,48,37,47,38,31)(26,40,34,42,43,45,44,46);(2,6)(7,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,32,33,21)(20,36,37,38)(25,41,30,39)(26,34,43,44)(31,47,48,35)(40,46,45,42)

Polycyclic presentation:

2^9|a^2=gh,c^2=fgi,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7499413);