#453497

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 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)(9,10,15,19)(11,21,25,37)(12,26)(13,28)(14,20,32,18)(16,33,36,24)(17,29)(22,30)(23,40)(27,31)(34,35)(38,39);(3,7)(5,8)(9,11,14,24,15,25,32,33)(10,16,18,21,19,36,20,37)(12,22,27,40,29,39,28,35)(13,30,17,34,31,38,26,23);(1,3,4,7)(2,5,6,8)(9,12)(10,17)(11,22)(13,20)(14,27)(15,29)(16,34)(18,31)(19,26)(21,38)(23,36)(24,40)(25,39)(28,32)(30,37)(33,35);(9,13,14,26,15,31,32,17)(10,12,20,27,19,29,18,28)(11,23,24,38,25,34,33,30)(16,35,37,22,36,40,21,39)

Polycyclic presentation:

2^9|a^2=efgh,b^2=e,c^2=g,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 360693);