#452796

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

830 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,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, 359992);