#8622177

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(17,18,24,26)(19,20,22,23)(21,25,30,32)(27,28,29,31)(34,37)(35,43)(36,44)(42,47);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(17,19)(18,20)(21,29)(22,24)(23,26)(25,31)(27,30)(28,32)(33,34)(35,40)(36,45)(37,39)(38,42)(41,44)(43,46)(47,48);(3,8)(5,13)(11,14)(12,16)(17,20,24,23)(18,19,26,22)(21,28,30,31)(25,27,32,29)(33,35,39,43)(34,40,37,46)(36,38,44,48)(41,47,45,42);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,31)(24,30)(26,32)(33,36)(34,41)(35,38)(37,45)(39,44)(40,47)(42,46)(43,48);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,22,24,19)(18,20,26,23)(21,29,30,27)(25,28,32,31)(33,37,39,34)(35,40,43,46)(36,45,44,41)(38,47,48,42);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,23,24,20)(18,19,26,22)(21,28,30,31)(25,29,32,27)(33,38)(34,42)(35,44)(36,43)(37,47)(39,48)(40,41)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,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, 8529373);