#7165838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#60931	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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)(17,18,24,30)(19,31,20,34)(21,27,39,45)(22,29,23,28)(25,36,26,33)(32,47,35,43)(37,42,48,41)(38,46,40,44);(3,9)(4,12)(6,13)(7,11)(17,19,24,20)(18,25,30,26)(21,37,39,48)(22,36,23,33)(27,43,45,47)(28,34,29,31)(32,40,35,38)(41,46,42,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,24,19)(18,26,30,25)(21,35,39,32)(22,36,23,33)(27,42,45,41)(28,34,29,31)(37,38,48,40)(43,44,47,46);(3,9)(6,13)(10,15)(14,16)(17,21)(18,27)(19,32)(20,35)(22,40)(23,38)(24,39)(25,41)(26,42)(28,46)(29,44)(30,45)(31,47)(33,48)(34,43)(36,37);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7073034);