#8757116

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	#46076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664312);