#7558969

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7466165);