#8197808

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8105004);