#7696410

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603606);