#7184008

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	#535	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1558 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=fi,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, 7091204);