#8688357

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

5141 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,15,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30);(2,3)(6,7)(10,14)(11,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,15)(14,16);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31);(1,4)(2,6)(3,7)(5,8)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30);(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595553);