#212495

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^c=dh,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, 119691);