#123671

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(5,6,7,8)(9,10)(11,16)(12,23)(13,26)(14,18)(15,19)(17,24)(20,36)(21,32)(22,31)(25,35)(27,30)(28,34)(29,37)(33,38)(39,40);(1,2)(3,4)(9,11,14,21,15,22,13,27)(10,16,18,32,19,31,26,30)(12,20,24,38,25,39,28,37)(17,33,35,40,34,29,23,36);(1,3,4,2)(9,12)(10,17)(11,20)(13,28)(14,24)(15,25)(16,29)(18,35)(19,34)(21,38)(22,39)(23,26)(27,37)(30,40)(31,33)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30867);