#8103902

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	#54465	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=g,e^2=g,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, 8011098);