#123558

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#392
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2538 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)(11,21)(12,26)(13,20)(14,18)(15,19)(16,23)(17,28)(22,33)(24,32)(25,31)(27,35)(29,34)(30,39)(36,37)(38,40);(2,4)(9,11,14,24,15,25,13,23)(10,16,20,31,19,32,18,21)(12,22,28,38,29,39,27,37)(17,33,26,36,35,30,34,40);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,26)(21,36)(23,37)(24,38)(25,39)(31,40)(32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^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,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, 30754);