#478409

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=i,e^2=i,g^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,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei

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, 385605);