#108992

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16188);