#149109

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	#1924
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

446 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:

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

Polycyclic presentation:

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

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, 56305);