#8755063

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(8,11)(10,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(1,2,5,6)(3,7,9,13)(4,11,12,8)(10,16,15,14)(20,28)(23,30)(26,32)(29,31)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(3,9)(7,13)(10,15)(14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662259);