#452170

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1192 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:

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

Polycyclic presentation:

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

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, 359366);