#8696689

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

5084 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603885);