#10444930

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

4902 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)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,25)(21,31)(22,27)(24,30)(26,28)(29,32);(2,4)(6,7)(10,11)(14,15)(17,19,22,29)(18,23,27,32)(20,28,30,21)(24,31,25,26);(9,11,13,10)(12,15,16,14)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,3,5,8)(2,6,4,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,27)(31,32);(1,4,5,2)(3,6,8,7)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(17,21,22,28)(18,26,27,31)(19,20,29,30)(23,24,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10352126);