#449750

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60423	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1487 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:

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

Polycyclic presentation:

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

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, 356946);