#146485

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,e^c=ei,f^c=fi,f^d=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, 53681);