#474313

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

569 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)(3,8)(4,6)(5,7)(9,10)(11,15,19,28)(12,22,23,32)(13,18)(14,17)(16,24,31,26)(20,35)(21,30,34,38)(25,27)(29,37)(33,40)(36,39);(9,11,13,21,14,19,25,34)(10,15,27,38,17,28,18,30)(12,20,26,37,23,33,24,36)(16,29,22,35,31,39,32,40);(1,3,5,4)(2,6,7,8)(11,19)(13,25)(15,28)(16,31)(18,27)(24,26)(29,39)(36,37);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,26)(14,23)(15,29)(17,31)(18,32)(19,33)(21,37)(22,27)(24,25)(28,39)(30,40)(34,36)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381509);