#7782712

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

4776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689908);