#147823

order	512
nilpotency class	4
isoclinism	isoclinic class of #129633
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1059 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:

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

Polycyclic presentation:

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

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, 55019);