#7791908

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

4679 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7699104);