#8181595

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

5382 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=g,e^2=h,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, 8088791);