#8811194

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gh,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, 8718390);