#108863

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

702 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)(5,6,8,10,7,9,11,12)(13,14)(15,24,27,20)(16,29)(17,33)(18,23)(19,22)(21,30)(25,39,41,43)(26,35,28,38)(31,36)(32,44)(34,40,37,42);(2,4)(13,15,18,28,19,27,17,26)(14,20,23,35,22,24,33,38)(16,25,31,42,32,41,30,40)(21,34,29,43,36,37,44,39);(6,9)(10,12)(13,16,17,30,19,32,18,31)(14,21,23,29,22,36,33,44)(15,25,26,40,27,41,28,42)(20,34,35,43,24,37,38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=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, 16059);