#8463711

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

4839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370907);