#6408892

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,d^c=di,e^d=ei

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, 6316088);