#10166166

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

4071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gh,e^2=hi,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, 10073362);