#108177

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15373);