#8158781

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065977);