#9851004

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	#51956	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=h,e^2=i,f^2=g,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, 9758200);