#6440839

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348035);