#123982

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1441
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1546 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:

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

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,c^b=cf,d^b=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, 31178);