#467675

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

694 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,21)(13,18,31,36)(14,19)(16,20)(17,26)(22,30,35,29)(23,25)(24,34,39,40)(27,33)(28,32,38,37);(1,3,5,4)(2,6,7,8)(9,11,15,10,16,19,14,20)(12,23,26,33,27,17,25,21)(13,22,29,18,31,35,30,36)(24,37,28,40,39,32,38,34);(2,7)(6,8)(9,12,16,27)(10,17,20,23)(11,21,19,33)(13,28,31,38)(14,26,15,25)(18,34,36,40)(22,37,35,32)(24,30,39,29);(1,4,5,3)(2,8,7,6)(9,13)(10,18)(11,22)(12,24)(14,30)(15,29)(16,31)(17,32)(19,35)(20,36)(21,34)(23,37)(25,38)(26,28)(27,39)(33,40)

Polycyclic presentation:

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

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, 374871);