#317338

order	512
nilpotency class	3
isoclinism	isoclinic class of #317315
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3214	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224534);