#8438586

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	#40670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8345782);