#10144535

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

4823 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gi,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, 10051731);