#504752

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

380 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)(9,10,13,17)(11,19,21,15)(12,24,26,16)(14,18,27,33)(20,29,37,35)(22,30,38,36)(23,34,28,31)(25,40,39,32);(1,3,5,4)(2,6,7,8)(9,11,13,21)(10,15,17,19)(12,20,26,37)(14,28,27,23)(16,29,24,35)(18,34,33,31)(22,39,38,25)(30,40,36,32);(1,4,5,3)(2,6,7,8)(11,20)(12,25)(14,27)(15,29)(16,32)(18,33)(19,35)(21,37)(22,23)(24,40)(26,39)(28,38)(30,31)(34,36);(2,7)(6,8)(9,12,14,25,13,26,27,39)(10,16,18,32,17,24,33,40)(11,20,28,22,21,37,23,38)(15,29,34,30,19,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411948);