#8436774

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	#49061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343970);