#8159197

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	#53135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(3,7,9,4)(5,13,14,10)(11,12,16,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(20,28)(25,30)(27,29)(31,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(2,8)(3,9)(10,13)(11,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,15)(12,13)(14,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,e^2=gh,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, 8066393);