#7002219

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

1396 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)(19,30)(20,34)(21,27)(22,28)(23,29)(24,39)(25,33)(26,35)(31,45)(32,43)(36,41)(37,44)(38,42)(40,47)(46,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,36)(22,38)(23,33)(24,32)(26,40)(27,45)(28,43)(29,30)(34,41)(35,48)(37,47)(39,42)(44,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,44)(19,31,33,36)(22,35,24,37)(25,40,30,46)(27,39,34,28)(32,47,38,48)(41,43,45,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,27,29,34)(19,31,33,36)(22,35,24,37)(25,41,30,45)(26,39,44,28)(32,48,38,47)(40,43,46,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,38)(20,35,21,37)(25,42,30,43)(26,27,44,34)(31,47,36,48)(40,41,46,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=f,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, 6909415);