#123691

order	512
nilpotency class	3
isoclinism	isoclinic class of #105681
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1434
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30887);