#7125202

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	#61185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1641 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,24,29)(19,30,34,44)(20,35,32,46)(21,27,31,42)(22,28,23,41)(25,40,45,33)(26,37,43,36)(38,47,39,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,22,40,24,34,23,33)(18,25,41,44,29,45,28,30)(20,21,36,39,32,31,37,38)(26,27,35,47,43,42,46,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,24,32)(18,26,29,43)(19,31,34,21)(22,36,23,37)(25,42,45,27)(28,46,41,35)(30,47,44,48)(33,39,40,38);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,27)(19,32)(20,34)(22,38)(23,39)(24,31)(25,43)(26,45)(28,47)(29,42)(30,46)(33,37)(35,44)(36,40)(41,48);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,28,29,41)(19,33,34,40)(20,36,32,37)(21,38,31,39)(25,44,45,30)(26,46,43,35)(27,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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, 7032398);