#7492926

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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)(19,24,22,37)(20,30,33,39)(21,26,36,44)(25,28,41,31)(29,38,34,42)(32,47,35,48)(40,45,43,46);(2,5)(3,7)(10,14)(11,12)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,34,36,29)(25,39,41,30)(26,42,44,38)(32,45,35,46)(40,47,43,48);(9,10)(11,16)(12,15)(13,14)(17,20,19,28,23,33,22,31)(18,25,37,39,27,41,24,30)(21,35,29,46,36,32,34,45)(26,43,42,47,44,40,38,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,32)(22,34)(23,36)(24,38)(25,40)(27,44)(28,45)(30,47)(31,46)(33,35)(37,42)(39,48)(41,43);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,19,23,22)(18,24,27,37)(20,28,33,31)(21,34,36,29)(25,30,41,39)(26,42,44,38)(32,46,35,45)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7400122);