#109451

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,22,26,23,27,21,37)(19,28,32,42,30,24,31,39)(20,33,35,40,36,48,34,25)(29,43,46,38,44,47,45,41);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,24,27,28)(20,29)(21,31)(22,32)(23,30)(25,38,40,41)(26,39,37,42)(33,47,48,43)(34,45)(35,46)(36,44);(3,10)(6,14)(9,15)(11,16)(17,20,21,34,23,36,22,35)(18,25,26,33,27,40,37,48)(19,29,31,45,30,44,32,46)(24,38,39,47,28,41,42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237230, #72237232, #72237233, …
maximal normal subgroup of	#72237230, #72237232, #72237233, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16647);