#8183844

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	#39006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(18,21)(19,27)(20,30)(26,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(17,18,23,21)(19,28,27,24)(20,31,30,25)(22,26,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,20,23,30)(18,25,21,31)(19,29,27,22)(24,32,28,26)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,21,23,18)(19,28,27,24)(20,31,30,25)(22,32,29,26)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,22,23,29)(18,26,21,32)(19,30,27,20)(24,31,28,25)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091040);