#7283551

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7190747);