#393098

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300294);