#7125198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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	#3484	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

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, 7032394);