#7124133

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	#61276	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)(19,28)(20,34)(21,38)(22,26)(23,31)(24,36)(25,37)(27,39)(29,41)(30,45)(32,42)(33,43)(35,44)(40,48)(46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,22,33)(18,23,26,42)(20,29,37,48)(21,32,39,31)(24,40,44,41)(25,45,34,46)(27,43,38,28)(30,36,47,35);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,29)(21,35)(22,37)(23,40)(25,27)(26,44)(28,46)(30,32)(31,47)(33,48)(34,38)(36,39)(41,42)(43,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,41)(24,27)(26,34)(28,46)(29,32)(31,48)(33,47)(35,39)(38,44)(40,42)(43,45);(2,8)(6,9)(7,16)(14,15)(18,26)(23,42)(24,44)(25,34)(27,38)(28,43)(40,41)(45,46)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,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, 7031329);