#112204

order	512
nilpotency class	3
isoclinism	isoclinic class of #112199
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

777 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:

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

Polycyclic presentation:

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

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, 19400);