#289394

order	512
nilpotency class	3
isoclinism	isoclinic class of #289363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196590);