#7124136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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	#61270	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 7031332);