#7684937

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

4784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592133);