#7649936

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

4882 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,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, 7557132);