#7254963

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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)(32,43)(33,38)(34,41)(35,46)(39,44)(42,48)(45,47);(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)(21,29,34,35)(25,38,40,30)(26,37,41,46)(32,44,45,42)(39,47,48,43);(9,11)(10,13)(12,16)(14,15)(17,20,22,31,23,33,19,28)(18,25,24,30,27,40,36,38)(21,32,35,44,34,45,29,42)(26,39,37,43,41,48,46,47);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,37)(25,39)(27,41)(28,42)(30,43)(31,44)(33,45)(36,46)(38,47)(40,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(21,34)(26,41)(29,35)(32,45)(37,46)(39,48)(42,44)(43,47)

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=df,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, 7162159);