#418989

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2273 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)(9,10,14,20,16,21,15,19)(11,22,25,17,27,35,26,36)(12,28,30,38,13,32,31,18)(23,40,29,37,24,34,33,39);(1,2,3,4)(9,11,14,25,16,27,15,26)(10,17,19,22,21,36,20,35)(12,29,30,24,13,33,31,23)(18,37,32,40,38,39,28,34);(6,8)(9,12,16,13)(10,18,21,38)(11,23,27,24)(14,31,15,30)(17,34,36,40)(19,28,20,32)(22,39,35,37)(25,33,26,29);(2,4)(9,13,16,12)(10,18,21,38)(11,24,27,23)(14,30,15,31)(17,34,36,40)(19,28,20,32)(22,39,35,37)(25,29,26,33)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=i,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,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 326185);