#7645553

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

4976 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552749);