#9945517

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	#40796	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852713);