#8437842

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

6112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,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, 8345038);