#8437795

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,21)(19,24,26,23)(20,25,29,28)(27,31,32,30)(33,34,39,38)(35,40,46,45)(36,42,37,41)(43,48,44,47);(2,3)(7,9)(8,10)(14,15)(19,26)(23,24)(27,32)(30,31)(33,35,39,46)(34,40,38,45)(36,43,37,44)(41,47,42,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,31)(29,32)(33,36,39,37)(34,41,38,42)(35,43,46,44)(40,47,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,29)(18,25,21,28)(19,27,26,32)(23,30,24,31)(33,37,39,36)(34,42,38,41)(35,44,46,43)(40,48,45,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,18)(19,24,26,23)(20,28,29,25)(27,31,32,30)(33,38,39,34)(35,45,46,40)(36,42,37,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8344991);