#114534

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

856 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)(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20)(21,22)(23,32,36,47)(24,29)(25,30)(26,41)(27,31)(28,34,43,35)(33,49,51,44)(37,52,50,42)(38,45)(39,48)(40,46);(6,10)(8,15)(14,19)(18,20)(21,23)(22,28)(24,37)(25,35)(26,34)(27,36)(29,44)(30,47)(31,43)(32,41)(33,39)(38,51)(40,50)(42,45)(46,49)(48,52);(2,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,27,40)(22,29,31,46)(23,33,36,51)(25,39,26,38)(28,42,43,52)(30,48,41,45)(32,49,47,44)(34,37,35,50)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21730);