#7236194

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(17,18,23,27)(19,24,22,36)(20,30,33,37)(21,26,35,42)(25,32,40,28)(29,38,34,41)(31,44,45,47)(39,46,48,43);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,35,34)(25,37,40,30)(26,38,42,41)(31,43,45,46)(39,47,48,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,40)(19,28,22,32)(21,31,35,45)(24,37,36,30)(26,39,42,48)(29,43,34,46)(38,47,41,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21)(18,26)(19,29)(20,31)(22,34)(23,35)(24,38)(25,39)(27,42)(28,43)(30,44)(32,46)(33,45)(36,41)(37,47)(40,48);(2,5)(6,8)(10,14)(13,15)(17,19,23,22)(18,24,27,36)(20,32,33,28)(21,34,35,29)(25,30,40,37)(26,41,42,38)(31,43,45,46)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7143390);