#114491

order	512
nilpotency class	3
isoclinism	isoclinic class of #114482
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21687);