#8901442

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

3927 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=g,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, 8808638);