#8649018

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

3879 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)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,28)(21,22)(23,24)(25,31)(27,30)(29,32)(33,34,39,36)(35,40,43,37)(38,41,46,44)(42,47,48,45);(1,2)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,19,23,22)(18,21,24,26)(20,29,31,27)(25,32,28,30)(33,35)(34,40)(36,37)(38,42)(39,43)(41,47)(44,45)(46,48);(2,4)(7,9)(8,12)(15,16)(18,24)(21,26)(25,28)(30,32)(33,34,39,36)(35,37,43,40)(38,41,46,44)(42,45,48,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,23,31)(18,25,24,28)(19,27,22,29)(21,30,26,32)(33,36,39,34)(35,37,43,40)(38,41,46,44)(42,47,48,45);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,30)(22,24)(23,26)(25,27)(28,29)(31,32)(33,37)(34,35)(36,43)(38,45)(39,40)(41,42)(44,48)(46,47);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,21,24,26)(20,27,31,29)(25,32,28,30)(33,38,39,46)(34,41,36,44)(35,42,43,48)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=gh,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, 8556214);