#229194

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

593 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:

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

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,f^a=fh,d^b=dei,e^b=eh,g^b=gh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136390);