#7086275

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6993471);