#8810134

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

4162 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)(19,24)(20,21)(22,23)(25,28)(26,32)(27,29)(30,31)(33,34,40,44)(35,37,38,41)(36,43,39,42)(45,46,48,47);(1,2,6,3)(4,9,11,7)(5,10,13,8)(12,14,16,15)(18,23)(21,31)(24,25)(27,32)(33,35)(34,41)(36,45)(37,44)(38,40)(39,48)(42,46)(43,47);(2,3)(4,11)(8,10)(12,16)(17,19,22,28)(18,24,23,25)(20,29,30,26)(21,27,31,32)(33,36,40,39)(34,42,44,43)(35,45,38,48)(37,47,41,46);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19)(18,25)(20,26)(21,27)(22,28)(23,24)(29,30)(31,32)(33,37,40,41)(34,35,44,38)(36,46,39,47)(42,48,43,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,21)(19,26)(22,30)(23,31)(24,32)(25,27)(28,29)(33,38,40,35)(34,41,44,37)(36,45,39,48)(42,47,43,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,20)(19,27)(22,31)(23,30)(24,29)(25,26)(28,32)(33,39,40,36)(34,43,44,42)(35,45,38,48)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=h,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, 8717330);