#148680

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
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

532 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:

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

Polycyclic presentation:

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

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, 55876);