#231484

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138680);