#7167152

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(18,23)(19,27)(21,37)(25,39)(26,40)(29,45)(30,31)(32,44)(33,47)(34,35)(38,42)(46,48);(3,9)(4,12)(6,13)(7,11)(17,18,21,23,22,26,37,40)(19,28,30,42,31,24,27,38)(20,32,34,44,36,39,35,25)(29,46,45,41,33,48,47,43);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,38)(25,43)(26,28)(27,37)(29,36)(32,46)(34,47)(35,45)(39,48)(40,42)(41,44);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,39)(24,41)(26,44)(27,45)(28,43)(30,47)(31,33)(32,40)(34,37)(38,46)(42,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,26)(23,40)(24,28)(25,44)(32,39)(38,42)(41,43)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,e^2=hi,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, 7074348);