#7011708

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,37)(19,24,32,41,33,42,31,48)(20,34,35,44,36,26,38,25)(29,40,45,43,30,46,47,39);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,43)(26,40)(27,41)(28,42)(29,36)(34,46)(35,47)(37,48)(38,45)(39,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,36)(18,25,28,44)(19,29,33,30)(21,35,22,38)(24,39,42,43)(26,37,34,27)(31,45,32,47)(40,41,46,48);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,38)(22,35)(23,36)(24,40)(25,27)(28,34)(29,33)(31,45)(32,47)(37,44)(39,48)(41,43)(42,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,35,36,38)(24,41,42,48)(25,26,44,34)(29,45,30,47)(39,46,43,40)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6918904);