#193704

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100900);