#8898526

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

4090 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)(4,13)(5,8)(6,10)(9,12)(11,16)(14,15)(17,18,24,28)(19,29,32,25)(20,23,22,26)(21,31,30,27)(33,34,38,41)(35,39,45,42)(36,37,40,46)(43,44,47,48);(1,3)(2,7)(4,14)(5,11)(6,12)(8,16)(9,10)(13,15)(17,19,24,32)(18,25,28,29)(20,30,22,21)(23,31,26,27)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,4,6,5)(2,8,10,13)(3,11,12,14)(7,15,9,16)(17,20,24,22)(18,26,28,23)(19,30,32,21)(25,27,29,31)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(1,5,6,4)(2,8,10,13)(3,11,12,14)(7,16,9,15)(17,21)(18,27)(19,20)(22,32)(23,25)(24,30)(26,29)(28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,2)(3,9)(4,8)(5,13)(6,10)(7,12)(11,16)(14,15)(17,22,24,20)(18,26,28,23)(19,30,32,21)(25,31,29,27)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48);(1,3)(2,9)(4,14)(5,11)(6,12)(7,10)(8,15)(13,16)(17,23,24,26)(18,20,28,22)(19,31,32,27)(21,25,30,29)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8805722);