#7717358

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

5140 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)(10,13)(14,16)(17,18)(19,26)(20,24)(21,31)(22,32)(23,25)(27,30)(28,29);(9,10,12,13)(11,14,15,16)(17,19,23,20)(18,24,25,26)(21,29,32,27)(22,28,31,30);(2,5)(6,7)(18,25)(19,20)(21,32)(28,30);(1,2)(3,6)(4,5)(7,8)(17,20,23,19)(18,24,25,26)(21,29,32,27)(22,30,31,28);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,30)(25,31)(26,28);(17,22)(18,21)(19,28)(20,30)(23,31)(24,29)(25,32)(26,27)

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7624554);