#147680

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239852, #72239857, #72239866, …
maximal normal subgroup of	#72239852, #72239857, #72239866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54876);