#7523437

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2232 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,26)(21,27)(22,28)(23,29)(25,32)(30,38)(31,39)(33,42)(34,43)(35,44)(36,41)(37,45)(40,47)(46,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,36)(25,28)(26,42)(27,39)(29,41)(30,34)(35,46)(37,47)(38,43)(40,45)(44,48);(10,14)(11,15)(17,19)(18,25)(20,30)(21,36)(22,32)(23,31)(24,28)(26,40)(27,39)(29,41)(33,34)(35,47)(37,46)(38,44)(42,45)(43,48);(10,14)(13,16)(17,20,22,35)(18,26,28,44)(19,30,32,47)(21,34,23,37)(24,38,25,40)(27,43,29,45)(31,46,36,33)(39,48,41,42);(2,6)(5,7)(17,21,22,23)(18,27,28,29)(19,31,32,36)(20,34,35,37)(24,39,25,41)(26,43,44,45)(30,46,47,33)(38,48,40,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=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, 7430633);