#10457958

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

5250 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=g,e^2=g,f^2=h,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, 10365154);