#117817

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 25013);