#8109192

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

4906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8016388);