#398056

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 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)(11,15)(12,24)(13,29)(14,19)(16,27)(17,22)(18,26)(20,33)(21,32)(23,31)(25,37)(28,39)(30,35)(34,40)(36,38);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,25)(13,22)(14,23)(16,30)(17,29)(18,32)(19,31)(20,28)(21,26)(24,37)(27,35)(33,39)(34,38)(36,40);(1,4,5,3)(2,6,7,8)(9,12,26,40,14,28,13,27)(10,16,29,39,19,34,18,24)(11,20,22,36,23,25,21,35)(15,30,32,37,31,38,17,33);(9,11)(10,17)(12,20)(13,21)(14,23)(15,29)(16,33)(18,31)(19,32)(22,26)(24,38)(25,28)(27,35)(30,39)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,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, 305252);