#7167148

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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, 7074344);