#8534664

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

4098 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)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(2,7)(3,10)(8,15)(11,13)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(34,39)(35,42)(41,44)(43,48);(2,7)(4,9)(8,15)(12,16)(18,25)(19,21)(26,32)(27,29)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=i,f^2=g,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, 8441860);