#418915

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326111);