#7558992

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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)(17,18,21,25)(19,27,30,23)(20,24,32,37)(22,26,34,39)(28,41,43,35)(29,36,31,40)(33,38,45,47)(42,46,44,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,27)(26,40)(31,34)(32,43)(33,42)(36,39)(37,41)(38,48)(44,45)(46,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,21,32)(18,24,25,37)(19,28,30,43)(22,33,34,45)(23,35,27,41)(26,38,39,47)(29,42,31,44)(36,46,40,48);(9,11,13,12)(10,16,14,15)(18,25)(19,29)(22,34)(23,36)(24,37)(27,40)(28,42)(30,31)(33,45)(35,46)(41,48)(43,44);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(18,25)(19,30)(20,32)(26,39)(29,31)(33,45)(35,41)(46,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7466188);