#7019151

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926347);