#9851089

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	#47894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 9758285);