#10108664

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	#38521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,11)(4,14)(5,10)(6,8)(7,9)(12,16)(13,15)(17,18,23,24)(19,21,26,31)(20,25,29,32)(22,30,28,27)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(1,3,7,6)(2,8,9,11)(4,12,10,15)(5,16,14,13)(17,19)(18,21)(20,28)(22,29)(23,26)(24,31)(25,27)(30,32)(33,35,39,46)(34,40,43,44)(36,45,37,38)(41,42,47,48);(2,9)(4,10)(8,11)(12,15)(18,24)(19,26)(20,29)(27,30)(33,36,39,37)(34,41,43,47)(35,45,46,38)(40,42,44,48);(1,4)(2,5)(3,12)(6,15)(7,10)(8,16)(9,14)(11,13)(17,20,23,29)(18,25,24,32)(19,28,26,22)(21,27,31,30)(33,37,39,36)(34,41,43,47)(35,38,46,45)(40,42,44,48);(1,5)(2,10)(3,13)(4,9)(6,16)(7,14)(8,12)(11,15)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,28)(27,29)(33,34)(35,40)(36,41)(37,47)(38,48)(39,43)(42,45)(44,46);(1,6,7,3)(2,11,9,8)(4,15,10,12)(5,13,14,16)(17,22,23,28)(18,27,24,30)(19,29,26,20)(21,25,31,32)(33,38)(34,42)(35,36)(37,46)(39,45)(40,47)(41,44)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10015860);