#148580

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1409 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55776);