#298217

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,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, 205413);