#263391

order	512
nilpotency class	3
isoclinism	isoclinic class of #263387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=di,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170587);