#8393684

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

4138 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,24)(19,27,30,22)(20,31,29,25)(21,32,28,26);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,23,30)(18,22,24,27)(20,28,29,21)(25,26,31,32)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(4,12)(7,14)(10,16)(13,15)(18,24)(22,27)(25,31)(26,32)(34,39)(37,45)(40,41)(43,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,25)(19,28)(21,30)(22,26)(23,29)(24,31)(27,32)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,21,23,28)(18,26,24,32)(19,29,30,20)(22,25,27,31)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,27)(18,19,24,30)(20,26,29,32)(21,25,28,31)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300880);