#474306

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

645 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)(3,6)(4,5)(7,8)(9,10,13,20)(11,21)(12,26,29,33)(14,17,15,19)(16,25)(18,27,34,28)(22,31)(23,37)(24,32)(30,38)(35,36)(39,40);(2,5)(6,8)(9,11,14,22,13,25,15,24)(10,16,19,31,20,21,17,32)(12,23,27,36,29,39,28,38)(18,30,33,40,34,35,26,37);(2,5)(3,7)(10,17)(11,22)(14,15)(16,21)(18,33)(19,20)(23,38)(24,25)(26,34)(27,28)(30,35)(36,39);(1,3,4,7)(2,6,5,8)(9,12,14,27,13,29,15,28)(10,18,19,33,20,34,17,26)(11,23,22,36,25,39,24,38)(16,30,31,40,21,35,32,37)

Polycyclic presentation:

2^9|a^2=efg,b^2=efgh,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, 381502);