#7659528

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

4579 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,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, 7566724);