#7167158

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	#60834	C₂
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)(17,18)(19,31)(20,37)(21,27)(22,29)(23,28)(24,30)(25,35)(26,39)(32,44)(33,47)(34,43)(36,42)(38,41)(40,45)(46,48);(3,9)(4,12)(6,13)(7,11)(17,19,23,36,24,34,22,35)(18,25,29,43,30,42,28,31)(20,38,40,48,21,32,39,33)(26,44,27,46,45,41,37,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,26)(19,32)(21,24)(22,39)(23,40)(25,41)(27,29)(28,37)(30,45)(31,46)(33,36)(34,38)(35,48)(42,44)(43,47);(3,9)(6,13)(10,15)(14,16)(17,21,24,20)(18,27,30,37)(19,33,34,48)(22,39,23,40)(25,41,42,44)(26,28,45,29)(31,47,43,46)(32,35,38,36);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,34)(25,42)(31,43)(32,38)(33,48)(35,36)(41,44)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,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, 7074354);