#8496515

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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)(3,9)(4,13)(5,6)(7,8)(10,16)(11,12)(14,15)(20,28)(24,29)(27,32)(30,31)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,8)(5,16)(9,13)(11,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(34,38)(35,42)(40,44)(43,48);(1,3,7,4)(2,9,8,13)(5,11,16,14)(6,12,10,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,7,3)(2,9,8,13)(5,14,16,11)(6,12,10,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,5)(2,10)(3,11)(4,14)(6,8)(7,16)(9,15)(12,13)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6,7,10)(2,5,8,16)(3,12,4,15)(9,11,13,14)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403711);