#474258

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
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

646 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efgh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,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, 381454);