#8622138

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529334);