#7204219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,30)(21,34)(22,37)(23,27)(25,28)(26,36)(29,42)(31,41)(32,46)(33,38)(35,39)(40,45)(43,48)(44,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,35,36,29)(25,30,41,38)(26,42,34,39)(32,43,47,45)(40,46,48,44);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,32,36,47)(24,38,37,30)(26,40,34,48)(29,43,35,45)(39,46,42,44);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,21)(18,26)(19,29)(20,32)(22,35)(23,36)(24,39)(25,40)(27,34)(28,43)(30,44)(31,45)(33,47)(37,42)(38,46)(41,48);(3,7)(5,8)(10,15)(14,16)(17,19,23,22)(18,24,27,37)(20,28,33,31)(21,35,36,29)(25,38,41,30)(26,42,34,39)(32,45,47,43)(40,44,48,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7111415);