#147331

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	#1889
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,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, 54527);