#7566806

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,26)(19,23,28,37)(20,30,31,40)(21,33,35,25)(24,34,39,32)(27,41,43,48)(29,45,44,38)(36,46,47,42);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,28)(18,23,26,37)(20,27,31,43)(21,29,35,44)(24,36,39,47)(25,38,33,45)(30,41,40,48)(32,42,34,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,40)(26,39)(28,43)(29,46)(30,33)(32,35)(37,47)(38,48)(41,45)(42,44);(2,6)(7,8)(10,14)(13,15)(19,28)(20,31)(23,37)(24,39)(29,44)(30,40)(32,34)(38,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=fg,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,e^d=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, 7474002);