#10166175

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	#50026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=h,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, 10073371);