#147335

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1867
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=i,g^2=i,b^a=bd,c^a=ce,h^a=hi,e^b=ehi,d^c=dh,e^d=ei

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, 54531);