#8193698

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

4949 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=ghi,e^2=ghi,f^2=g,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, 8100894);