#113340

order	512
nilpotency class	3
isoclinism	isoclinic class of #113339
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

778 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)(4,8)(5,7)(10,13,16,12)(14,15)(17,18)(19,27,30,40)(20,31,33,24)(21,25)(22,26)(23,29,37,34)(28,42)(32,45,47,39)(35,41)(36,46)(38,43)(44,48);(1,3,5,11)(2,6,7,9)(4,13,15,12)(8,10,14,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,38)(25,40)(26,37)(27,41)(29,35)(31,36)(32,46)(33,44)(39,42)(43,47)(45,48);(1,2)(3,10)(4,14)(5,7)(6,13)(8,15)(9,12)(11,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,39)(26,31)(27,42)(29,43)(30,44)(34,46)(35,47)(37,38)(40,48)(41,45)

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=fi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20536);