#8118934

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026130);