#123925

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1436
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1550 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)(5,6,9,15,11,16,10,14)(7,17,19,12,21,28,20,29)(8,22,24,13,25,31,26,32)(18,27,23,36,34,35,33,30);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,28,16,29,14,17)(8,23,24,34,25,33,26,18)(13,30,22,35,32,36,31,27);(5,8,9,24,11,25,10,26)(6,13,14,22,16,32,15,31)(7,18,20,33,21,34,19,23)(12,27,28,36,29,35,17,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,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, 31121);