#10166256

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18,22,24)(19,26,29,23)(20,28,30,25)(21,31,32,27)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19,22,29)(18,23,24,26)(20,27,30,31)(21,28,32,25)(34,40)(35,37)(42,46)(43,45);(2,9)(3,4)(10,14)(11,12)(18,24)(20,30)(23,26)(27,31)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,26,32,23)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,19)(18,26)(20,31)(21,25)(22,29)(23,24)(27,30)(28,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,21)(18,27)(19,28)(20,26)(22,32)(23,30)(24,31)(25,29)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,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, 10073452);