#395326

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

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

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, 302522);