#7728259

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

5048 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(18,22)(19,26)(24,28)(27,32);(1,3,5,7)(2,4,6,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,7)(3,6)(5,8)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7635455);