#7336636

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=f,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, 7243832);