#425614

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1261 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)(3,6)(4,8)(5,7)(9,10)(11,22)(12,27)(13,21)(14,20)(15,19)(16,25)(17,30)(18,31)(23,39)(24,34)(26,36)(28,33)(29,37)(32,35)(38,40);(1,3,5,4)(2,6,7,8)(9,11,14,26)(10,16,20,35)(12,28,30,40)(13,32,15,25)(17,34,27,39)(18,38,37,33)(19,36,21,22)(23,29,24,31);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,23)(13,29)(14,30)(15,31)(16,33)(18,21)(19,37)(20,27)(22,34)(24,26)(25,40)(28,32)(35,38)(36,39);(9,12)(10,18)(11,24)(13,31)(14,30)(15,29)(16,34)(17,21)(19,27)(20,37)(22,33)(23,26)(25,40)(28,32)(35,39)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332810);