#114077

order	512
nilpotency class	3
isoclinism	isoclinic class of #114076
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

783 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:

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

Polycyclic presentation:

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

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, 21273);