#6404516

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61316	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311712);