#10481078

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

5183 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)(9,10,13,15)(11,12,16,14)(17,18,23,20)(19,27,29,24)(21,25,28,30)(22,26,32,31);(9,11,13,16)(10,14,15,12)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,31,30,26);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,14)(15,16)(17,20,23,18)(19,24,29,27)(21,30,28,25)(22,31,32,26);(1,3)(2,6)(4,7)(5,8)(17,21)(18,25)(19,22)(20,30)(23,28)(24,31)(26,27)(29,32);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,25)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388274);