#7901965

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

4699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809161);