#9203782

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

5340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9110978);