#110419

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 17615);