#8157340

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

5068 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)(3,7,8,4)(9,10,13,11)(12,15,16,14)(17,18,23,25)(19,24,20,26)(21,29,31,27)(22,30,32,28);(1,3)(2,4)(5,8)(6,7)(10,11)(14,15)(17,19)(18,24)(20,23)(21,28)(22,29)(25,26)(27,32)(30,31);(1,4)(2,3)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(18,25)(19,20)(27,29)(28,30);(2,6)(4,7)(9,12)(10,14)(11,15)(13,16)(17,20,23,19)(18,26,25,24)(21,28,31,30)(22,27,32,29);(1,4)(2,3)(5,7)(6,8)(17,21,23,31)(18,27,25,29)(19,28,20,30)(22,24,32,26);(1,3)(2,4)(5,8)(6,7)(17,22)(18,28)(19,29)(20,27)(21,24)(23,32)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8064536);