#7799521

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

4745 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=i,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, 7706717);