#363199

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

2452 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)(11,20,24,15)(12,25,27,17)(13,19)(14,18)(16,26,34,28)(21,36)(22,31)(23,37,40,32)(29,35)(30,38)(33,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,23)(14,24)(16,33)(17,31)(18,20)(19,32)(21,27)(25,36)(26,39)(28,38)(29,40)(30,34)(35,37);(2,5)(6,8)(9,12)(10,16)(11,21)(13,28)(14,27)(15,30)(17,19)(18,34)(20,33)(22,24)(23,39)(25,35)(26,29)(31,37)(32,36)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,31)(16,19)(18,25)(20,36)(21,24)(23,39)(28,29)(30,37)(32,33)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=h,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,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, 270395);