#363067

order	512
nilpotency class	3
isoclinism	isoclinic class of #190034
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,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, 270263);