#8093341

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

4902 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000537);