#7726438

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

5012 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)(17,18,23,25)(19,28,20,24)(21,22,27,26)(29,30,31,32);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,24,25,28)(21,29,27,31)(22,32,26,30);(2,6)(3,7)(9,12,13,10)(11,15,16,14)(18,25)(19,20)(21,27)(30,32);(17,20,23,19)(18,24,25,28)(21,31,27,29)(22,32,26,30);(1,3,5,7)(2,4,6,8)(17,21,23,27)(18,26,25,22)(19,29,20,31)(24,30,28,32);(1,4)(2,7)(3,6)(5,8)(17,22,23,26)(18,27,25,21)(19,30,20,32)(24,29,28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7633634);