#10363655

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

5088 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10270851);