#8455969

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	#37496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=g,e^2=gi,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, 8363165);