#8746783

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,22,21)(19,23,26,24)(20,25,28,29)(27,30,32,31)(34,38)(35,41)(40,43)(42,44);(2,7)(3,8)(9,16)(10,13)(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,32)(29,31)(34,38)(39,46)(40,43)(47,48);(2,7)(3,8)(9,16)(10,13)(19,26)(23,24)(27,32)(30,31)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,21,22,18)(19,24,26,23)(20,25,28,29)(27,30,32,31)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653979);