#7492958

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#53078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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)(20,30,33,38)(21,26,35,42)(25,28,40,31)(29,37,34,41)(32,44,46,47)(39,43,48,45);(2,5)(3,7)(10,14)(11,12)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,34,35,29)(25,38,40,30)(26,41,42,37)(32,43,46,45)(39,44,48,47);(9,10)(11,16)(12,15)(13,14)(17,20,22,31,23,33,19,28)(18,25,24,30,27,40,36,38)(21,32,34,45,35,46,29,43)(26,39,37,44,42,48,41,47);(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)(23,35)(24,37)(25,39)(27,42)(28,43)(30,44)(31,45)(33,46)(36,41)(38,47)(40,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,19,23,22)(18,24,27,36)(20,28,33,31)(21,34,35,29)(25,30,40,38)(26,41,42,37)(32,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,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=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400154);