#7166247

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	#60928	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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)(17,18,22,26)(19,28,33,41)(20,25,36,43)(21,37,38,27)(23,32,40,31)(24,34,42,35)(29,46,48,39)(30,45,47,44);(3,9)(4,12)(6,13)(7,11)(17,19)(18,23)(20,29)(21,32)(22,33)(24,39)(25,44)(26,40)(27,41)(28,37)(30,34)(31,38)(35,47)(36,48)(42,46)(43,45);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,24)(19,29)(21,34)(22,36)(23,39)(25,37)(26,42)(27,43)(28,44)(30,32)(31,47)(33,48)(35,38)(40,46)(41,45);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,39)(24,37)(26,43)(27,42)(28,45)(29,32)(31,48)(33,47)(34,38)(40,46)(41,44);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,26)(23,40)(24,42)(25,43)(27,37)(28,41)(39,46)(44,45)

Polycyclic presentation:

2^9|a^2=hi,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,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, 7073443);