#9835854

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	#56307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18,23,21)(19,27,29,24)(20,30,28,25)(22,26,32,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,6)(7,14)(8,11)(15,16)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,28)(18,25,21,30)(19,22,29,32)(24,26,27,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,18)(19,24,29,27)(20,30,28,25)(22,26,32,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9743050);