#231217

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 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)(11,18)(12,15)(13,17)(14,22)(16,20)(19,24)(21,23)(25,26,30,33,31,35,29,34)(27,32,36,39,37,40,28,38);(1,2)(3,6)(4,5)(7,8)(11,19)(14,23)(18,24)(21,22)(25,27,30,36,31,37,29,28)(26,32,33,39,35,40,34,38);(1,3,4,7)(2,6,5,8)(9,11,20,21,13,19,12,22)(10,14,15,24,17,23,16,18)(25,28,29,37,31,36,30,27)(26,32,33,39,35,40,34,38);(2,5)(6,8)(9,12,13,20)(10,15,17,16)(11,21,19,22)(14,18,23,24)(25,29,31,30)(26,33,35,34)(27,28,37,36)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=h,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, 138413);