#6400611

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	#61303	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(17,18,22,26,23,27,21,36)(19,28,31,24,29,40,37,38)(20,25,35,43,32,41,34,48)(30,44,45,39,33,46,47,42);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(19,29)(20,32)(24,38)(25,41)(28,40)(31,37)(34,35)(43,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,24)(20,33)(21,37)(22,31)(23,29)(25,42)(26,40)(27,38)(28,36)(30,32)(34,45)(35,47)(39,41)(43,44)(46,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,20,22,35,23,32,21,34)(18,25,26,43,27,41,36,48)(19,30,31,45,29,33,37,47)(24,39,40,46,38,42,28,44)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,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^b=ei,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307807);