#146693

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303107, #72303122, #72303183, …
maximal normal subgroup of	#72303107, #72303122, #72303183, …
maximal quotient of	#72303634

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53889);