#10168478

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=ghi,f^2=gh,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, 10075674);