#9945524

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

4721 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)(3,5)(4,6)(7,8)(10,13)(15,16)(18,22)(20,27)(23,29)(25,28);(1,2,4,6)(3,5,8,7)(9,10,14,13)(11,16,12,15)(18,22)(23,29)(24,30)(31,32);(9,10,14,13)(11,16,12,15)(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32);(1,2)(3,7)(4,6)(5,8)(9,12,14,11)(10,15,13,16)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31);(1,3,4,8)(2,5,6,7)(9,13,14,10)(11,16,12,15)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,28)(29,32)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852720);