#10166316

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

5021 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)(3,4)(5,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30);(1,3,6,5)(2,7,8,4)(10,11)(14,15)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27);(9,11,13,10)(12,15,16,14)(18,25)(19,29)(20,31)(27,32);(1,4,6,7)(2,5,8,3)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32);(1,5,6,3)(2,7,8,4)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24);(1,2)(3,7)(4,5)(6,8)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073512);