#8093645

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

5234 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:

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

Polycyclic presentation:

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

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, 8000841);