#146492

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

854 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:

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

Polycyclic presentation:

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

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, 53688);