#7590001

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1624 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)(17,18,22,26)(19,23,29,36)(20,24,31,38)(21,25,33,40)(27,34,43,47)(28,35,30,37)(32,39,44,48)(41,45,42,46);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,34)(25,35)(26,36)(30,33)(31,43)(32,41)(37,40)(38,47)(39,45)(42,44)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,31)(18,24,26,38)(19,27,29,43)(21,32,33,44)(23,34,36,47)(25,39,40,48)(28,41,30,42)(35,45,37,46);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,28)(21,33)(23,35)(25,40)(27,41)(29,30)(32,44)(34,45)(36,37)(39,48)(42,43)(46,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,29)(20,31)(23,36)(24,38)(28,30)(32,44)(35,37)(39,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=fg,e^2=fg,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, 7497197);