#374610

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1428 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)(3,7)(4,6)(5,8)(9,10,15,20,13,18,14,19)(11,21,24,35,23,16,30,32)(12,25,29,17,27,34,28,33)(22,37,38,31,26,40,39,36);(1,3,5,4)(2,6,8,7)(9,11,13,23)(10,16,18,21)(12,26,27,22)(14,30,15,24)(17,31,33,36)(19,35,20,32)(25,37,34,40)(28,38,29,39);(1,3,5,4)(2,7,8,6)(9,12,13,27)(10,17,18,33)(11,22)(14,29,15,28)(16,31)(19,34,20,25)(21,36)(23,26)(24,39)(30,38)(32,37)(35,40);(1,4,5,3)(2,7,8,6)(10,18)(16,21)(17,33)(19,20)(25,34)(31,36)(32,35)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281806);