#205964

order	512
nilpotency class	3
isoclinism	isoclinic class of #205777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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)(3,6)(4,8)(5,7)(9,10,13,18,14,17,19,33)(11,15,22,28,21,29,26,34)(12,16,25,32,24,31,35,38)(20,27,36,39,23,30,37,40);(1,2,6,8,5,7,4,3)(9,11,19,22,14,21,13,26)(10,15,18,34,17,29,33,28)(12,23,35,37,24,20,25,36)(16,30,32,39,31,27,38,40);(1,3,6,2,5,8,4,7)(9,12,14,24)(10,16,17,31)(11,20,21,23)(13,25,19,35)(15,27,29,30)(18,32,33,38)(22,36,26,37)(28,39,34,40);(12,24)(16,31)(20,23)(25,35)(27,30)(32,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113160);