#147336

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
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

697 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)(5,6,10,15,11,16,25,32)(7,17,8,13,19,29,21,30)(9,14,24,31,23,28,34,36)(12,20,27,33,26,22,35,18);(5,7,10,8,11,19,25,21)(6,12,15,27,16,26,32,35)(9,18,24,20,23,33,34,22)(13,28,29,36,30,14,17,31);(1,3,4,2)(5,8,10,19,11,21,25,7)(6,13,15,29,16,30,32,17)(9,22,24,18,23,20,34,33)(12,14,27,31,26,28,35,36)

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,c^b=ci,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312047, #72312053, #72312075, …
maximal normal subgroup of	#72312047, #72312053, #72312075, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54532);