#8496837

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
prime radical	#42532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,23,22)(19,24,29,28)(20,30,21,25)(26,32,27,31)(33,34,40,38)(35,44,39,41)(36,43,37,42)(45,47,46,48);(2,4)(7,9)(8,12)(15,16)(17,19,23,29)(18,24,22,28)(20,27,21,26)(25,32,30,31)(33,35,40,39)(34,41,38,44)(36,45,37,46)(42,47,43,48);(17,20,23,21)(18,25,22,30)(19,26,29,27)(24,31,28,32)(33,36,40,37)(34,42,38,43)(35,45,39,46)(41,47,44,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,21,23,20)(18,25,22,30)(19,27,29,26)(24,31,28,32)(33,37,40,36)(34,43,38,42)(35,45,39,46)(41,47,44,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32)(33,38,40,34)(35,41,39,44)(36,43,37,42)(45,47,46,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,18)(19,28,29,24)(20,30,21,25)(26,32,27,31)(33,39,40,35)(34,41,38,44)(36,46,37,45)(42,47,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8404033);