#353020

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260216);