#10166301

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	#50368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gh,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, 10073497);