#8551218

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

4081 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,38)(35,40,42,37)(36,41,45,46)(43,47,48,44);(2,7)(3,9)(10,15)(11,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,42)(34,40,38,37)(36,43,45,48)(41,47,46,44);(2,7)(4,8)(10,15)(12,16)(21,29)(24,31)(26,32)(28,30)(35,42)(37,40)(43,48)(44,47);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,35)(36,44)(38,42)(39,40)(41,43)(45,47)(46,48);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,38,39,34)(35,37,42,40)(36,41,45,46)(43,47,48,44)

Polycyclic presentation:

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

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, 8458414);