#229185

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

631 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 136381);