#123728

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#397
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,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, 30924);