#8425566

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

4855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332762);