#7657897

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

5140 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565093);