#212523

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

545 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26,30,28,29,32,37,36)(27,31,35,34,33,38,40,39);(1,3,5,4)(2,6,7,8)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(27,33)(28,36)(35,40);(1,4,5,3)(2,6,7,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,28,30,32,29,36,37,26)(27,34,35,38,33,39,40,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=egh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dh,d^c=di,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, 119719);