#8751692

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

5012 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658888);