#10087740

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

4832 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=gh,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, 9994936);