#7283553

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(17,18,23,24,22,27,19,28)(20,31,29,40,33,37,36,25)(21,26,35,38,34,41,30,42)(32,44,43,48,45,47,46,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,36)(21,30,34,35)(25,37,40,31)(26,38,41,42)(32,43,45,46)(39,47,48,44);(10,13)(11,15)(17,20,22,33)(18,25,27,40)(19,29,23,36)(21,32,34,45)(24,37,28,31)(26,39,41,48)(30,43,35,46)(38,47,42,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,38)(25,39)(27,41)(28,42)(29,43)(31,44)(33,45)(36,46)(37,47)(40,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,27)(21,34)(24,28)(25,40)(30,35)(31,37)(32,45)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,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, 7190749);