#126849

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 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)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(21,22)(23,34,37,50)(24,29)(25,42)(26,45)(27,33)(28,36,47,43)(30,40)(31,44)(32,41)(35,48,39,52)(38,46,51,49);(2,4)(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,23,26,38,41,35,44,36,27,37,40,51,24,39,25,43)(22,28,31,48,29,49,45,34,33,47,42,52,32,46,30,50);(7,15)(11,19)(14,20)(16,17)(21,24,27,41)(22,29,33,32)(23,35,37,39)(25,40,44,26)(28,46,47,49)(30,31,45,42)(34,48,50,52)(36,51,43,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=chi,d^b=di,e^b=ei

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, 34045);