#8456804

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

5367 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31);(3,4)(6,8)(10,12)(14,15)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,15)(13,16);(1,4,5,3)(2,6,7,8)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31);(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8364000);