#8110472

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	#45616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017668);