#7566822

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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)(19,27)(20,31)(23,28)(24,39)(30,42)(32,40)(33,36)(38,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,25)(19,28,27,23)(20,32,31,40)(22,35,34,26)(24,36,39,33)(29,45,41,48)(30,38,42,44)(37,46,43,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,29)(21,27)(22,30)(24,37)(25,28)(26,38)(31,41)(32,48)(33,46)(34,42)(35,44)(36,47)(39,43)(40,45);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,21,31)(18,24,25,39)(19,29,27,41)(22,36,34,33)(23,37,28,43)(26,40,35,32)(30,47,42,46)(38,45,44,48);(2,6)(7,8)(10,14)(13,15)(19,27)(20,31)(23,28)(24,39)(30,42)(32,40)(33,36)(38,44)

Polycyclic presentation:

2^9|a^2=gi,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, 7474018);