#7019065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,25,21,27)(19,28,31,24,29,32,20,30);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,31)(22,29)(25,30)(26,28)(27,32);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(18,25)(19,29)(20,31)(26,27);(11,16)(13,15)(17,20,23,19,22,31,21,29)(18,24,26,32,25,30,27,28);(10,14)(13,15)(17,21,22,23)(18,26,25,27)(19,20,29,31)(24,32,30,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

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, 6926261);