#8701947

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

5220 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8609143);