#146973

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54169);