#7167149

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	#60833	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1676 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)(19,29)(20,34)(21,26)(22,40)(23,27)(24,32)(25,36)(28,30)(31,41)(33,47)(35,43)(37,44)(38,46)(39,45)(42,48);(3,9)(4,12)(6,13)(7,11)(17,19,30,48,23,33,22,32)(18,24,40,47,27,42,28,29)(20,31,36,38,37,21,35,39)(25,41,34,45,43,26,44,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,25)(19,31)(21,33)(22,35)(23,37)(24,41)(26,42)(27,43)(28,44)(29,46)(30,36)(32,39)(34,40)(38,48)(45,47);(3,9)(6,13)(10,15)(14,16)(17,21,23,31)(18,26,27,41)(19,20,33,37)(22,39,30,38)(24,25,42,43)(28,46,40,45)(29,34,47,44)(32,36,48,35);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,42)(25,43)(26,41)(28,40)(29,47)(34,44)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=i,e^2=h,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, 7074345);