#108979

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,e^2=i,f^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, 16175);