#6410414

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

1223 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)(19,27)(20,26)(21,25)(22,24)(23,28)(29,31)(30,32)(33,34,39,42,38,43,37,47)(35,44,45,48,46,40,36,41);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,30)(21,29)(22,28)(24,27)(25,32)(26,31)(33,35)(34,40)(36,37)(38,46)(39,45)(41,42)(43,44)(47,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(18,24)(19,28)(25,26)(29,30)(33,36,38,45)(34,41,43,48)(35,37,46,39)(40,42,44,47);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,25,24,26)(19,29,28,30)(23,31,27,32)(33,37,38,39)(34,42,43,47)(35,45,46,36)(40,48,44,41);(2,10)(7,11)(8,9)(15,16)(17,21,22,20)(18,26,24,25)(19,29,28,30)(23,31,27,32)(34,43)(35,46)(36,45)(42,47)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di,e^c=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, 6317610);