#7198401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fgh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105597);