#106524

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1776 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)(4,6)(5,8)(9,10,14,18,15,19,13,27)(11,20,24,30,23,35,22,16)(12,17,25,32,26,33,28,31)(21,36,37,29,38,34,39,40);(1,3)(2,5,6,7)(4,8)(9,11)(10,16,19,30)(12,21)(13,22)(14,24)(15,23)(17,29,33,40)(18,20,27,35)(25,37)(26,38)(28,39)(31,36,32,34);(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,29)(18,32)(19,33)(20,34)(22,39)(23,38)(24,37)(27,31)(30,40)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=eh,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280834, #72281113, #72281137
maximal normal subgroup of	#72280834, #72281113, #72281137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13720);