#146473

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	C₂×#342
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1738 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)(7,15,17,27)(8,19,20,12)(9,22)(10,14)(11,16)(13,23)(18,32,31,28)(21,26,34,33)(24,36)(25,30)(29,35);(5,7,9,18,10,17,23,31)(6,11,13,24,14,25,22,35)(8,16,21,29,20,30,34,36)(12,15,26,28,19,27,33,32);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,f^b=fhi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308038, #72308075, #72308082
maximal normal subgroup of	#72308038, #72308075, #72308082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53669);