#298220

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

989 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205416);