#8810135

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

4180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717331);