#363380

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1037 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=g,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^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, 270576);