#363216

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2448 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,f^c=fi,h^d=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, 270412);