#472328

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

866 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)(4,5)(7,8)(9,10,15,19)(11,20,25,23)(12,17,27,33)(13,21,18,14)(16,30,32,24)(22,36,38,35)(26,34,29,28)(31,39,40,37);(2,5)(6,8)(9,11,21,30,15,25,14,24)(10,16,13,23,19,32,18,20)(12,22,26,37,27,38,29,39)(17,31,28,36,33,40,34,35);(3,7)(6,8)(9,12)(10,17)(11,22,25,38)(13,28)(14,29)(15,27)(16,31,32,40)(18,34)(19,33)(20,35,23,36)(21,26)(24,39,30,37);(1,3,4,7)(2,6,5,8)(9,13,14,10,15,18,21,19)(11,23,24,16,25,20,30,32)(12,17,26,28,27,33,29,34)(22,31,37,36,38,40,39,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=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, 379524);