#7338691

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7245887);