#115520

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
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

905 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22716);