#474228

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
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

520 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)(3,5)(4,6)(7,8)(9,10)(11,22,25,36)(12,17,24,31)(13,18)(14,19)(15,21)(16,20)(23,32,38,40)(26,30,37,39)(27,33)(28,35)(29,34);(1,3,4,7)(2,5,6,8)(9,11)(10,17,20,31)(12,15)(13,23)(14,24)(16,25)(18,30,34,39)(19,22,21,36)(26,28)(27,37)(29,38)(32,35,40,33);(9,12,14,25,16,24,15,11)(10,17,21,22,20,31,19,36)(13,23,27,26,29,38,28,37)(18,32,35,39,34,40,33,30);(2,6)(5,8)(9,13,14,27,16,29,15,28)(10,18,19,33,20,34,21,35)(11,23,24,37,25,38,12,26)(17,30,22,32,31,39,36,40)

Polycyclic presentation:

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

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, 381424);