#474307

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,20)(12,26,29,28)(13,15,32,16)(17,25)(18,31,36,30)(21,38)(22,23)(24,34)(27,33)(35,40)(37,39);(2,5)(6,8)(9,11,16,24,14,25,15,23)(10,17,32,34,19,20,13,22)(12,27,31,40,29,39,30,21)(18,35,28,33,36,38,26,37);(2,5)(3,7)(9,12,14,29)(10,18,19,36)(11,21,25,40)(13,28,32,26)(15,31,16,30)(17,33,20,37)(22,38,34,35)(23,27,24,39);(1,3,4,7)(2,6,5,8)(9,13,15,19,14,32,16,10)(11,22,23,20,25,34,24,17)(12,28,30,18,29,26,31,36)(21,35,39,37,40,38,27,33)

Polycyclic presentation:

2^9|a^2=efg,b^2=efgh,c^2=i,d^2=f,f^2=i,g^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,e^b=ei,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi

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, 381503);