#148672

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,f^c=fi,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, 55868);