#108406

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gh,f^b=fh,f^c=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15602);