#364438

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2396 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=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, 271634);