#126670

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

442 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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22,24,28,25,29,34,40,26,30,35,41,36,42,46,50)(23,27,32,38,33,39,45,49,31,37,43,47,44,48,51,52);(2,4)(5,7,9,17,10,18,8,16)(6,11,12,19,14,20,13,15)(21,23)(22,27)(24,32)(25,33)(26,31)(28,38)(29,39)(30,37)(34,45)(35,43)(36,44)(40,49)(41,47)(42,48)(46,51)(50,52);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(23,31)(27,37)(32,43)(33,44)(38,47)(39,48)(45,51)(49,52)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33866);