#8826280

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

4150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733476);