#6400652

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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

1591 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)(17,18,22,26,23,27,21,35)(19,28,31,24,29,39,36,37)(20,25,34,41,32,40,33,45)(30,42,44,38,43,48,46,47);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(19,29)(24,37)(28,39)(30,43)(31,36)(38,47)(42,48)(44,46);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,29)(25,38)(26,39)(27,37)(28,35)(32,43)(33,46)(34,44)(40,47)(41,48)(42,45);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(20,32)(25,40)(30,43)(33,34)(38,47)(41,45)(42,48)(44,46);(2,10)(7,11)(8,9)(15,16)(17,20,22,34,23,32,21,33)(18,25,26,41,27,40,35,45)(19,30,31,44,29,43,36,46)(24,38,39,48,37,47,28,42)

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,g^c=gi,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, 6307848);