#368049

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322821
maximal normal subgroup of	#72322821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275245);