#7806667

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

5021 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)(9,10,13,14)(11,12,15,16)(17,18,23,25)(19,24,27,32)(20,28,30,26)(21,31,29,22);(1,2,4,5)(3,6,7,8)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,32)(26,31);(10,14)(11,15)(18,25)(19,27)(20,30)(22,31);(9,11)(10,12)(13,15)(14,16)(17,20)(18,26)(19,21)(22,24)(23,30)(25,28)(27,29)(31,32);(9,12)(10,11)(13,16)(14,15)(17,21)(18,22)(19,20)(23,29)(24,26)(25,31)(27,30)(28,32);(1,3,4,7)(2,6,5,8)(17,22)(18,21)(19,26)(20,24)(23,31)(25,29)(27,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713863);