#105302

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

642 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=i,g^2=h,b^a=bd,d^a=dh,c^b=ci,e^b=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12498);