#8093246

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

4142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,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

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, 8000442);