#147981

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55177);