#10089140

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

5099 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=g,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, 9996336);