#7766500

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

5268 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,16,13,15)(17,18,23,20)(19,27,28,24)(21,25,31,29)(22,26,32,30);(9,11,14,13)(10,15,12,16)(17,19)(18,24)(20,27)(21,22)(23,28)(25,30)(26,29)(31,32);(2,5)(6,7)(10,12)(15,16)(19,28)(22,32)(24,27)(26,30);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,12,15)(17,21)(18,25)(19,22)(20,29)(23,31)(24,30)(26,27)(28,32);(17,22)(18,26)(19,21)(20,30)(23,32)(24,29)(25,27)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7673696);