#7336447

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fi,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=ehi,d^c=dh,e^c=ei,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, 7243643);