#451896

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=dei,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359092);