#148280

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
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

808 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)(6,12)(7,17)(8,22)(9,10)(13,28)(14,33)(15,26)(16,27)(18,30)(19,34)(20,21)(25,31)(29,35)(32,36);(5,6,8,12,11,16,22,27)(7,18,19,28,21,31,35,36)(9,14,23,26,10,15,24,33)(13,29,30,20,32,34,25,17);(5,7,11,21)(6,13,16,32)(8,19,22,35)(9,20,10,17)(12,25,27,30)(14,18,15,31)(23,34,24,29)(26,36,33,28)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=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, 55476);