#7659579

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

4993 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7566775);