#7001958

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

1407 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:

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

Polycyclic presentation:

2^9|a^2=i,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=dhi,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, 6909154);