#456500

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1143 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,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, 363696);