#369249

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 276445);