#231220

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=chi,d^b=de,d^c=dhi

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, 138416);