#7624406

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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	#2186	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1176 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,30)(20,26)(21,28)(22,37)(23,27)(24,32)(25,33)(29,39)(31,42)(34,41)(35,46)(36,40)(38,44)(43,48)(45,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,40)(26,42)(27,41)(28,30)(29,43)(35,48)(36,47)(37,45)(38,39)(44,46);(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,40)(26,44)(27,43)(28,45)(29,41)(30,37)(33,39)(35,47)(36,48)(42,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,21,23)(18,26,28,27)(19,31,32,34)(22,35,36,39)(24,41,30,42)(25,44,45,43)(29,37,46,40)(33,38,47,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,27,28,26)(19,31,32,34)(22,35,36,39)(24,42,30,41)(25,43,45,44)(29,40,46,37)(33,38,47,48)

Polycyclic presentation:

2^9|b^2=fi,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,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, 7531602);