#6672294

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

1601 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)(19,24)(20,32)(21,26)(22,27)(23,29)(25,35)(28,37)(30,41)(31,33)(34,46)(36,42)(38,43)(39,44)(40,45)(47,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,24)(20,29)(21,25)(22,38)(23,32)(26,35)(27,43)(28,33)(30,39)(31,37)(34,42)(36,46)(40,47)(41,44)(45,48);(17,20,23,24)(18,25,29,31)(19,26,32,28)(21,33,37,35)(22,34,40,41)(27,38,45,47)(30,42,46,44)(36,48,39,43);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,23,37)(18,26,29,28)(19,25,32,31)(20,33,24,35)(22,36,40,39)(27,42,45,44)(30,38,46,47)(34,48,41,43);(3,11)(6,9)(10,16)(14,15)(17,22)(18,27)(19,30)(20,34)(21,36)(23,40)(24,41)(25,38)(26,42)(28,44)(29,45)(31,47)(32,46)(33,48)(35,43)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,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, 6579490);