#7624452

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531648);