#7782747

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

5135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689943);