#7166240

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	#60932	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(19,28)(21,22)(26,38)(27,35)(29,43)(30,31)(32,42)(33,36)(37,40)(44,47)(45,46);(3,9)(4,12)(6,13)(7,11)(17,18)(19,25)(20,32)(21,24)(22,35)(23,27)(26,33)(28,40)(29,44)(30,37)(31,41)(34,38)(36,42)(39,45)(43,48)(46,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,31)(24,37)(26,39)(27,41)(32,44)(33,45)(34,46)(35,40)(36,43)(38,47)(42,48);(3,9)(6,13)(10,15)(14,16)(17,20)(18,26)(19,29)(21,33)(22,36)(23,34)(24,38)(25,39)(27,42)(28,43)(30,45)(31,46)(32,35)(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)(18,24,27,35)(19,30,31,28)(20,33,34,36)(25,37,41,40)(26,38,42,32)(29,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=h,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,f^b=fi,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, 7073436);