#113406

order	512
nilpotency class	3
isoclinism	isoclinic class of #113404
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

846 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)(3,9,11,6)(4,13)(5,7)(8,14)(10,15,12,16)(17,18)(19,27,30,40)(20,31,33,24)(21,25)(22,26)(23,29,38,34)(28,43)(32,41,35,39)(36,42)(37,44)(45,48)(46,47);(1,3,5,11)(2,6,7,9)(4,10,14,12)(8,15,13,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,40)(26,38)(27,42)(29,36)(31,47)(32,46)(33,45)(35,44)(39,43)(41,48);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,41)(26,31)(27,43)(29,46)(30,45)(32,36)(34,44)(38,47)(39,42)(40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,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, 20602);