#8946210

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

4118 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(17,18,23,21)(19,24,26,20)(22,29,30,25)(27,28,32,31)(33,34,39,43)(35,42,47,40)(36,41,37,38)(44,46,45,48);(2,7)(4,8)(9,13)(12,15)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,7)(4,8)(5,14)(10,16)(18,21)(19,26)(22,30)(28,31)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,15,13,12)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,21,23,18)(19,24,26,20)(22,29,30,25)(27,31,32,28)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(11,16)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8853406);