#148681

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
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:

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

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,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, 55877);