#6672634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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

1603 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)(19,23)(20,33)(21,25)(22,27)(24,36)(26,38)(28,42)(29,30)(31,40)(32,41)(34,47)(35,44)(37,43)(39,45)(46,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,23)(20,34)(21,29)(22,32)(24,42)(25,30)(26,40)(27,41)(28,36)(31,38)(33,47)(35,46)(37,45)(39,43)(44,48);(17,20,22,37)(18,24,27,44)(19,28,32,46)(21,35,38,36)(23,39,41,47)(25,33,26,43)(29,34,31,45)(30,42,40,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,38)(18,25,27,26)(19,29,32,31)(20,35,37,36)(23,30,41,40)(24,33,44,43)(28,34,46,45)(39,42,47,48);(3,11)(6,9)(10,16)(14,15)(17,18)(19,30)(20,24)(21,26)(22,27)(23,29)(25,38)(28,42)(31,41)(32,40)(33,36)(34,39)(35,43)(37,44)(45,47)(46,48)

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,e^d=ei,h^e=hi

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, 6579830);