#10166651

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	#56855	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=g,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, 10073847);