#429019

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

987 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,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, 336215);