#8182916

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

5047 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=gh,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, 8090112);