#7806817

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

5323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^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, 7714013);