#10020328

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

4975 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9927524);