#7336529

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,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, 7243725);