#7590121

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,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, 7497317);