#8533263

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	#58008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(7,13)(8,11)(14,16)(17,18,22,23)(19,20,27,25)(21,24,30,31)(26,28,29,32)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,22,23)(19,25,27,20)(21,24,30,31)(26,32,29,28)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(3,9)(7,13)(10,15)(14,16)(18,23)(20,25)(24,31)(28,32)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,30)(18,24,23,31)(19,26,27,29)(20,28,25,32)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440459);