#7746739

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

4878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653935);