#7649946

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

4881 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7557142);