#8901638

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

3910 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808834);