#7206912

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(10,14)(11,12)(17,18,22,26)(19,27,28,40)(20,31,32,24)(21,34,36,25)(23,35,37,30)(29,43,42,41)(33,39,47,46)(38,44,45,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(19,28)(20,32)(23,37)(24,31)(27,40)(30,35)(33,47)(39,46);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,29)(21,35)(22,28)(24,38)(25,40)(26,37)(27,34)(30,36)(31,45)(32,42)(33,48)(39,43)(41,46)(44,47);(2,6)(3,7)(9,11,13,12)(10,16,14,15)(17,20,22,32)(18,24,26,31)(19,29,28,42)(21,33,36,47)(23,38,37,45)(25,39,34,46)(27,41,40,43)(30,44,35,48);(3,7)(5,8)(9,12,13,11)(10,16,14,15)(19,28)(20,32)(23,37)(24,31)(27,40)(30,35)(33,47)(39,46)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fi,e^2=f,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,e^c=ei,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, 7114108);