#7689142

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

4841 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=g,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, 7596338);