#114016

order	512
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60953	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

769 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:

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

Polycyclic presentation:

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

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, 21212);