#148584

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1820
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

645 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55780);