#362829

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1240 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)(3,7)(4,6)(5,8)(9,10,16,21,14,22,15,20)(11,17,26,34,24,35,25,33)(12,27,29,36,13,19,30,18)(23,32,37,40,28,39,38,31);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,23)(14,24)(15,26)(16,25)(18,31)(19,32)(20,34)(21,33)(22,35)(27,39)(29,37)(30,38)(36,40);(9,12,14,13)(10,18)(11,23,24,28)(15,29,16,30)(17,31)(19,21)(20,27)(22,36)(25,37,26,38)(32,33)(34,39)(35,40);(2,6)(5,7)(9,13,14,12)(10,19,22,27)(11,23,24,28)(15,30,16,29)(17,32,35,39)(18,21,36,20)(25,37,26,38)(31,33,40,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270025);