#8434220

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

5148 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=g,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, 8341416);