#8446026

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

4967 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353222);