#7683300

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

4733 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590496);