#111680

order	512
nilpotency class	3
isoclinism	isoclinic class of #111678
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,e^c=eh,g^c=gh

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, 18876);