#114013

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, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6)(7,8,17,11)(9,14)(10,13)(12,18,16,19)(15,20)(21,22)(23,32,35,43)(24,36)(25,30)(26,41)(27,31)(28,34,44,40)(29,38)(33,45,50,42)(37,46)(39,47)(48,51,52,49);(2,4)(5,7,10,17)(6,11,13,8)(9,12,20,16)(14,19,15,18)(21,23)(22,28)(24,33)(25,40)(26,34)(27,35)(29,45)(30,43)(31,44)(32,41)(36,52)(37,51)(38,49)(39,50)(42,46)(47,48);(5,8)(6,12)(7,14)(9,19)(10,11)(13,16)(15,17)(18,20)(21,24,27,39)(22,29,31,46)(23,33,35,50)(25,38,26,37)(28,42,44,45)(30,47,41,36)(32,48,43,52)(34,51,40,49)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=hi,f^2=h,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, 21209);