#6672290

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	#3013	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 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)(19,29)(20,35)(21,28)(22,27)(23,26)(24,33)(25,37)(30,46)(31,41)(32,34)(36,40)(38,44)(39,43)(42,48)(45,47);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,22,33)(18,24,27,29)(20,36,38,47)(21,31,23,34)(25,42,43,46)(26,41,28,32)(30,39,48,37)(35,45,44,40);(17,20,22,38)(18,25,27,43)(19,30,33,48)(21,37,23,39)(24,40,29,45)(26,35,28,44)(31,47,34,36)(32,46,41,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,23)(18,26,27,28)(19,31,33,34)(20,37,38,39)(24,41,29,32)(25,35,43,44)(30,47,48,36)(40,42,45,46);(3,11)(6,9)(10,16)(14,15)(17,18)(19,32)(20,25)(21,26)(22,27)(23,28)(24,31)(29,34)(30,46)(33,41)(35,37)(36,45)(38,43)(39,44)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,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, 6579486);