#10106277

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

4252 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34)(35,37)(36,40)(38,42)(39,46)(41,47)(43,48)(44,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35,38,41)(34,37,42,47)(36,44,46,43)(39,48,40,45);(2,3)(7,9)(8,10)(14,15)(18,24)(20,28)(23,30)(25,31)(33,36,38,46)(34,39,42,40)(35,43,41,44)(37,45,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,36)(34,40)(35,44)(37,45)(38,46)(39,42)(41,43)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,37)(36,43)(38,41)(39,45)(40,48)(42,47)(44,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,37)(34,41)(35,42)(36,45)(38,47)(39,44)(40,43)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^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, 10013473);