#8182030

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

5014 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)(3,7)(4,5)(6,8)(18,19)(20,27)(24,26)(28,32);(1,3,6,5)(2,7,8,4)(9,10,13,14)(11,15,16,12)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31);(2,8)(3,5)(10,14)(12,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31);(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,4)(2,3)(5,8)(6,7)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,5,6,3)(2,4,8,7)(9,12)(10,11)(13,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8089226);