#105822

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1694 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)(11,19)(12,23,24,16)(13,26)(14,17)(15,21)(18,27)(20,28,33,31)(22,29)(25,30,37,36)(32,35)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,22,14,21,27,35)(10,15,18,29,17,19,26,32)(12,20,25,34,24,33,37,40)(16,28,30,38,23,31,36,39);(9,12)(10,16)(11,20)(13,25)(14,24)(15,28)(17,23)(18,30)(19,31)(21,33)(22,34)(26,36)(27,37)(29,38)(32,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224724, #72224728, #72224733
maximal normal subgroup of	#72224724, #72224728, #72224733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13018);