#8093484

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

4123 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)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,24)(20,25)(21,28)(22,23)(26,31)(27,30)(29,32)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(2,6)(7,9)(8,12)(14,15)(19,27)(20,29)(24,30)(25,32)(34,36)(35,37)(41,44)(43,45);(3,10)(7,9)(11,16)(14,15)(18,23)(19,27)(25,32)(26,28)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,24)(20,28)(21,25)(22,27)(23,30)(26,29)(31,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000680);