#8105281

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8012477);