#148168

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	#1889
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,h^c=hi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250315, #72257225, #72257231, …
maximal normal subgroup of	#72250315, #72257225, #72257231, …
maximal quotient of	#72264922, #72308075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55364);