#398055

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,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, 305251);