#123593

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#379
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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,d^b=di,g^b=gi,d^c=di,g^c=gi

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, 30789);