#7806675

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

5217 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^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, 7713871);