#146683

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1078 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)(7,16,19,32)(8,13,20,28)(9,14,23,12)(10,15)(11,17)(18,31)(21,27)(22,30)(24,36,29,34)(25,33)(26,35);(1,2,3,4)(5,7)(6,11)(8,17)(9,22)(10,19)(12,24)(13,16)(14,29)(15,25)(18,23)(20,33)(21,36)(26,31)(27,30)(28,32)(34,35);(6,12)(8,20)(9,23)(11,24)(13,27)(14,15)(16,30)(17,33)(18,22)(25,29)(26,28)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53879);