#148670

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 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)(5,6,9,16,10,14,23,30,11,15,24,31,8,20,22,33)(7,12,18,27,17,13,28,29,19,26,32,36,21,25,34,35);(1,2,4,3)(5,7)(6,12)(8,21)(9,18)(10,17)(11,19)(13,14)(15,26)(16,27)(20,25)(22,34)(23,28)(24,32)(29,30)(31,36)(33,35);(1,3,4,2)(5,7)(6,13)(8,21)(9,18)(10,17)(11,19)(12,20)(14,26)(15,25)(16,29)(22,34)(23,28)(24,32)(27,33)(30,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55866);