#6672293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3012	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579489);