#7184014

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	#2002	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei

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, 7091210);