#10480698

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

5042 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)(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,22)(21,28)(23,27)(25,30)(26,31)(29,32);(10,11)(14,15)(17,18)(19,24)(20,22)(21,25)(23,27)(26,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,32)(29,30);(3,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,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, 10387894);