#478508

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385704);