#7173844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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	#61202	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(22,39)(23,27)(24,32)(25,30)(28,40)(31,41)(33,42)(35,43)(36,48)(37,46)(38,45)(44,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,35)(21,36)(22,33)(23,31)(25,43)(26,44)(27,42)(28,41)(29,39)(30,37)(32,40)(34,45)(38,47)(46,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,22,30,23,36,40,47)(18,25,28,34,27,44,39,48)(19,21,33,38,31,35,32,37)(24,26,41,46,42,43,29,45);(3,9)(6,13)(10,15)(14,16)(17,21)(18,26)(19,30)(20,32)(22,38)(23,35)(24,34)(25,29)(27,43)(28,46)(31,47)(33,36)(37,40)(39,45)(41,44)(42,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,31)(20,36)(26,43)(28,39)(30,47)(32,33)(45,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=hi,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,e^c=ei,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, 7081040);