#7597976

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,30,33,25)(20,34,38,45)(21,40,32,42)(22,29,24,27)(26,39,44,37)(31,43,47,48)(35,41,36,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,28,30)(20,35,38,36)(21,22,32,24)(26,43,44,48)(27,42,29,40)(31,37,47,39)(34,46,45,41);(10,13)(14,15)(17,20,24,39,23,38,22,37)(18,26,29,45,28,44,27,34)(19,31,32,35,33,47,21,36)(25,41,42,43,30,46,40,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,24,19,23,32,22,33)(18,25,29,42,28,30,27,40)(20,36,39,31,38,35,37,47)(26,41,45,43,44,46,34,48);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,29)(19,32,33,21)(20,37,38,39)(25,42,30,40)(26,34,44,45)(31,35,47,36)(41,43,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7505172);