#8213920

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

5077 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,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, 8121116);