#511893

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250062, #72250084, #72250086
maximal normal subgroup of	#72250062, #72250084, #72250086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419089);