#7002214

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1403 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)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,25,33,43)(20,34,21,27)(22,28,23,38)(26,36,44,35)(30,45,31,40)(32,41,37,42)(39,46,48,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,30)(21,31)(22,37)(23,32)(24,33)(26,39)(27,40)(28,42)(29,43)(34,45)(35,47)(36,46)(38,41)(44,48);(9,11)(10,13)(12,16)(14,15)(17,20,24,21)(18,26,29,44)(19,30,33,31)(22,35,23,36)(25,39,43,48)(27,38,34,28)(32,46,37,47)(40,41,45,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,24,20)(18,27,29,34)(19,31,33,30)(22,35,23,36)(25,40,43,45)(26,38,44,28)(32,46,37,47)(39,41,48,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,32,33,37)(20,35,21,36)(25,41,43,42)(26,27,44,34)(30,46,31,47)(39,45,48,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fi,e^2=fi,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,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, 6909410);