#10158601

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

4949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,e^2=i,f^2=g,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, 10065797);