#7007370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914566);