#8505720

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	#42578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 8412916);