#8633447

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540643);