#146491

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
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	#1782
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

888 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)(8,21,23,13)(9,14)(10,15)(11,18,27,17)(12,22,29,36)(19,28)(20,32)(24,30)(25,35)(26,34,31,33);(1,2,3,4)(5,7)(6,11)(8,18)(9,19)(10,20)(12,28)(13,16)(14,26)(15,27)(17,23)(21,32)(22,34)(24,35)(25,29)(30,31)(33,36);(5,8)(6,12,15,29)(7,17)(9,22)(10,23)(11,25,27,28)(13,14,21,30)(16,31,32,26)(18,20)(19,33)(24,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302717, #72302837, #72302855
maximal normal subgroup of	#72302717, #72302837, #72302855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53687);