#7737557

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

5199 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:

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

Polycyclic presentation:

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

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, 7644753);