#113399

order	512
nilpotency class	3
isoclinism	isoclinic class of #113393
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

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,7,3,6,9)(4,12,14,11,16,13,15,8)(17,18)(19,28,32,24)(20,33)(21,38)(22,27)(23,26)(25,36)(29,46,47,41)(30,40,31,43)(34,39,48,45)(35,44)(37,42);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,15)(12,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,28)(27,43)(29,37)(33,39)(35,48)(36,47)(38,40)(42,46)(44,45);(1,4,7,16)(2,8,3,11)(5,15,6,14)(9,12,10,13)(17,20,23,35)(18,25,26,42)(19,29,32,47)(21,37,22,36)(24,39,28,45)(27,44,38,33)(30,34,31,48)(40,46,43,41)

Polycyclic presentation:

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

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, 20595);