#8695859

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

4982 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,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, 8603055);