#8505771

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(20,28)(24,29)(27,32)(30,31)(33,34,39,38)(35,43,45,40)(36,41,47,46)(37,42,44,48);(2,7)(4,14)(8,10)(11,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,34,39,38)(35,43,45,40)(36,46,47,41)(37,48,44,42);(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,35,39,45)(34,40,38,43)(36,44,47,37)(41,42,46,48);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,15,9,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36)(34,41)(35,44)(37,45)(38,46)(39,47)(40,42)(43,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,15)(13,16)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,37,39,44)(34,42,38,48)(35,36,45,47)(40,46,43,41);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,16,10,11)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,38,39,34)(35,43,45,40)(36,41,47,46)(37,48,44,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=i,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, 8412967);