#8127312

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	#56544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=gh,f^2=g,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^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, 8034508);