#7696405

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

5352 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=gh,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, 7603601);