#146838

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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)(5,6,11,7,10,15,28,20,12,17,29,16,9,14,27,19)(8,21,23,35,25,13,30,31,26,32,24,34,22,33,36,18);(1,2,4,3)(5,7,11,14,10,20,28,6,12,16,29,15,9,19,27,17)(8,18,24,21,25,35,36,13,26,31,23,32,22,34,30,33);(1,3,4,2)(5,8,9,22,12,26,10,25)(6,13,14,21,17,33,15,32)(7,18,19,34,16,31,20,35)(11,24,27,30,29,23,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54034);