#277933

order	512
nilpotency class	3
isoclinism	isoclinic class of #277931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185129);