#113911

order	512
nilpotency class	3
isoclinism	isoclinic class of #113905
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

823 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21107);