#483633

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3171	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dgi,e^a=eg,c^b=ci,d^b=di,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320988, #72322004, #72322062
maximal normal subgroup of	#72320988, #72322004, #72322062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390829);