#7726032

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

4800 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=g,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, 7633228);