#123674

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	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ 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:

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

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,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ei

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, 30870);