#8508883

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	#39185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=hi,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, 8416079);