#362737

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2294 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)(11,21,24,16)(12,18,13,27)(14,20)(15,19)(17,26,34,28)(22,31)(23,37,40,32)(25,36)(29,35)(30,38)(33,39);(1,2,4,6)(3,7,8,5)(9,11,15,24)(10,16,19,21)(12,25,13,22)(14,23,29,40)(17,33,34,30)(18,31,27,36)(20,32,35,37)(26,39,28,38);(1,3,4,8)(2,5,6,7)(9,12,15,13)(10,17,19,34)(11,22,24,25)(14,28,29,26)(16,30,21,33)(18,20,27,35)(23,39,40,38)(31,37,36,32);(9,13,15,12)(10,18,19,27)(11,22,24,25)(14,28,29,26)(16,31,21,36)(17,20,34,35)(23,38,40,39)(30,37,33,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 269933);