#8445664

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

4873 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)(3,7)(4,8)(5,6)(9,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,25,31,28);(2,6)(3,8)(10,14)(11,15)(17,19,22,29)(18,23,26,27)(20,28,30,25)(21,24,31,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,18)(19,27)(20,24)(21,28)(22,26)(23,29)(25,31)(30,32);(1,4)(2,3)(5,7)(6,8)(17,20,22,30)(18,24,26,32)(19,28,29,25)(21,23,31,27);(17,21)(18,25)(19,24)(20,23)(22,31)(26,28)(27,30)(29,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,e^2=gi,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, 8352860);