#7806791

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

4996 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^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, 7713987);