#7758312

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

4997 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7665508);