#123661

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1438 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)(7,16,19,31)(8,21,23,13)(9,14)(10,25)(11,15)(12,18,27,20)(17,30)(22,35,24,29)(26,34)(28,32)(33,36);(1,2)(3,4)(5,7,10,20,11,19,9,18)(6,12,14,16,15,27,25,31)(8,17,24,34,23,33,22,32)(13,28,29,36,21,26,35,30);(2,4)(5,8,9,22,11,23,10,24)(6,13,14,29,15,21,25,35)(7,17,18,32,19,33,20,34)(12,26,16,30,27,28,31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,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, 30857);