#450930

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358126);