#7489960

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#42679	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(3,6)(4,5)(7,8)(17,18,23,25,22,28,21,27)(19,24,35,40,29,39,31,37)(20,32,36,42,33,26,34,41)(30,44,46,47,43,38,45,48);(2,5)(3,7)(10,14)(11,12)(17,19)(18,24)(20,30)(21,31)(22,29)(23,35)(25,40)(26,38)(27,37)(28,39)(32,44)(33,43)(34,45)(36,46)(41,48)(42,47);(9,10)(11,16)(12,15)(13,14)(18,25)(19,29)(20,33)(21,23)(24,37)(26,42)(27,28)(32,41)(38,48)(39,40)(44,47)(45,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,33)(18,26,28,32)(19,30,29,43)(21,34,23,36)(24,38,39,44)(25,41,27,42)(31,45,35,46)(37,47,40,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,29)(20,33)(24,39)(26,32)(31,35)(34,36)(37,40)(41,42)

Polycyclic presentation:

2^9|a^2=gi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7397156);