#218461

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

596 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125657);