#8184756

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	#40006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

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

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, 8091952);