#6672643

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

1607 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)(17,18,22,21)(19,28,31,25)(20,33,36,44)(23,39,24,27)(26,38,35,37)(29,43,46,48)(30,40,42,32)(34,45,47,41);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,25)(20,34)(21,28)(22,31)(23,32)(24,40)(26,43)(27,30)(29,38)(33,41)(35,48)(36,47)(37,46)(39,42)(44,45);(17,20,22,36)(18,26,21,35)(19,29,31,46)(23,38,24,37)(25,41,28,45)(27,44,39,33)(30,43,42,48)(32,47,40,34);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,21)(25,28)(26,35)(27,39)(30,42)(33,44)(41,45)(43,48);(3,11)(6,9)(10,16)(14,15)(17,21,22,18)(19,30,31,42)(20,35,36,26)(23,39,24,27)(25,32,28,40)(29,43,46,48)(33,37,44,38)(34,41,47,45)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,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,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, 6579839);