#8097018

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

4807 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004214);