#454494

order	512
nilpotency class	3
isoclinism	isoclinic class of #454021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1395 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361690);