#8105593

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(4,5,15,6)(11,14,16,12)(17,18)(19,21)(20,28)(22,26)(23,25)(24,30)(27,29)(31,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,7,13)(2,8,9,10)(4,11,15,16)(5,12,6,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,4)(2,5)(3,11)(6,9)(7,15)(8,12)(10,14)(13,16)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(1,5,7,6)(2,4,9,15)(3,12,13,14)(8,11,10,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(35,42)(36,45)(38,47)(39,43);(3,13)(8,10)(11,16)(12,14)(18,26)(19,25)(24,28)(27,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,4,9,15)(3,14,13,12)(8,11,10,16)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,f^2=ghi,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, 8012789);