#123964

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	C₂²⋊#95
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

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

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, 31160);