#147341

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

697 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,e^b=ehi,d^c=dh,e^c=ei,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, 54537);