#8127241

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

4251 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)(3,7)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(17,18,22,24)(19,20,27,23)(21,29,30,25)(26,31,32,28)(33,34,38,42)(35,41,37,39)(36,40,46,44)(43,48,45,47);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,19,22,27)(18,23,24,20)(21,26,30,32)(25,31,29,28)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(2,8)(3,9)(4,13)(11,16)(18,24)(20,23)(25,29)(28,31)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,30)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,21,22,30)(18,25,24,29)(19,26,27,32)(20,28,23,31)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

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, 8034437);