#8581768

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

4203 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(2,7)(3,10)(8,15)(11,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(35,36)(39,42)(43,44)(47,48);(2,7)(4,9)(8,15)(12,16)(19,25)(22,29)(26,31)(30,32)(33,35,38,36)(34,39,41,42)(37,44,46,43)(40,48,45,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=i,e^2=ghi,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, 8488964);