#7726433

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

4900 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,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, 7633629);