#474309

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

641 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efgh,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 381505);