#8110556

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5164 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017752);