#278001

order	512
nilpotency class	3
isoclinism	isoclinic class of #277971
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#548
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

502 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:

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

Polycyclic presentation:

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

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, 185197);