#8919406

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	#46694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826602);