#113513

order	512
nilpotency class	3
isoclinism	isoclinic class of #113506
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#37550	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,e^c=ei,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, 20709);