#7173835

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	#61204	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)(17,18,24,30)(19,31,35,42)(20,36,38,48)(21,27,37,44)(22,29,23,28)(25,33,43,34)(26,41,45,32)(39,47,40,46);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,37)(21,38)(22,34)(23,33)(24,35)(26,44)(27,45)(28,31)(29,42)(30,43)(32,39)(36,46)(40,41)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,32,24,38,22,41)(18,26,29,36,30,45,28,48)(19,21,33,40,35,37,34,39)(25,27,42,47,43,44,31,46);(3,9)(6,13)(10,15)(14,16)(17,21)(18,27)(19,32)(20,34)(22,39)(23,40)(24,37)(25,36)(26,31)(28,46)(29,47)(30,44)(33,38)(35,41)(42,45)(43,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,32,38,41)(21,39,37,40)(25,42,43,31)(26,36,45,48)(27,46,44,47)

Polycyclic presentation:

2^9|a^2=h,c^2=g,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,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, 7081031);