#489237

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#61401	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396433);