#148825

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56021);