#10171830

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	#46537	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,30,31,32)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,32,25,30)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,30,27,32)(33,34,38,42)(35,43,44,39)(36,40,47,46)(37,41,45,48);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,28)(22,31)(23,26)(24,30)(27,32)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,29,27,31)(33,36,38,47)(34,40,42,46)(35,37,44,45)(39,48,43,41);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,30,25,32)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,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, 10079026);