#8639755

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,5,6)(3,7,11,9)(4,8,13,14)(10,15,12,16)(17,18)(19,26)(20,28)(21,22)(23,24)(25,31)(27,30)(29,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,2,5,6)(3,9,11,7)(4,8,13,14)(10,16,12,15)(17,19,23,22)(18,21,24,26)(20,29,31,27)(25,32,28,30)(34,39)(40,42)(41,46)(47,48);(2,6)(7,9)(8,14)(15,16)(18,24)(21,26)(25,28)(30,32)(33,35,38,36)(34,40,39,42)(37,43,45,44)(41,47,46,48);(1,3)(2,7)(4,12)(5,11)(6,9)(8,16)(10,13)(14,15)(17,20,23,31)(18,25,24,28)(19,27,22,29)(21,30,26,32)(33,36,38,35)(34,40,39,42)(37,43,45,44)(41,48,46,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,14)(17,21)(18,19)(20,30)(22,24)(23,26)(25,27)(28,29)(31,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(1,4,5,13)(2,8,6,14)(3,10,11,12)(7,15,9,16)(17,22,23,19)(18,21,24,26)(20,27,31,29)(25,32,28,30)(33,37,38,45)(34,41,39,46)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8546951);