#6669715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,25)(22,26)(23,28)(24,34)(27,37)(29,32)(30,33)(31,35)(36,43)(38,44)(39,47)(40,45)(41,46)(42,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,31)(18,20,28,35)(21,29,37,24)(22,38,41,42)(25,32,27,34)(26,44,46,48)(30,36,47,40)(33,43,39,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,25)(21,32)(22,39)(23,35)(26,38)(27,31)(28,29)(30,45)(33,41)(34,37)(36,48)(40,44)(42,46)(43,47);(2,8)(6,9)(7,14)(13,15)(17,21,23,37)(18,25,28,27)(19,29,31,24)(20,32,35,34)(22,36,41,40)(26,43,46,45)(30,38,47,42)(33,44,39,48);(3,11)(6,9)(10,16)(13,15)(17,22,23,41)(18,26,28,46)(19,30,31,47)(20,33,35,39)(21,36,37,40)(24,42,29,38)(25,43,27,45)(32,44,34,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,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,e^c=ei,g^c=gi,h^c=hi,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, 6576911);