#146419

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

884 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)(5,6,10,18,12,7,14,20,13,16,11,28,9,17,25,26)(8,19,22,34,21,30,36,32,23,33,27,35,24,15,29,31);(2,4)(5,7,11,20,12,16,25,28,13,17,10,26,9,6,14,18)(8,19,22,35,21,30,36,31,23,33,27,34,24,15,29,32);(5,8)(6,15)(7,19)(9,24)(10,27)(11,22)(12,21)(13,23)(14,29)(16,30)(17,33)(18,32)(20,35)(25,36)(26,34)(28,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341758
maximal normal subgroup of	#72341758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53615);