#8445792

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

5326 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)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18,24,22)(19,29,30,25)(20,27,21,26)(23,28,31,32);(2,3)(6,7)(11,15)(14,16)(17,19,24,30)(18,25,22,29)(20,23,21,31)(26,28,27,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,22,27)(19,23,30,31)(25,28,29,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,22,26)(19,23,30,31)(25,28,29,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22,24,18)(19,29,30,25)(20,27,21,26)(23,32,31,28);(17,23)(18,28)(19,21)(20,30)(22,32)(24,31)(25,27)(26,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=h,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, 8352988);