#278516

order	512
nilpotency class	3
isoclinism	isoclinic class of #278491
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

529 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)(3,11)(4,7)(5,14)(6,8)(9,12)(10,15)(13,16)(17,18,22,27,21,26,23,28)(19,29,32,44,31,24,38,40)(20,33,36,42,35,43,37,25)(30,46,34,39,47,41,48,45);(1,3,5,4,6,12,15,13)(2,7,10,11,8,16,14,9)(17,19,21,31)(18,24,26,29)(20,34,35,48)(22,32,23,38)(25,41,42,46)(27,40,28,44)(30,36,47,37)(33,45,43,39);(2,8)(3,12)(4,13)(10,14)(17,20,21,35)(18,25,26,42)(19,30,31,47)(22,37,23,36)(24,39,29,45)(27,43,28,33)(32,48,38,34)(40,46,44,41);(1,4,5,12,6,13,15,3)(2,9,10,7,8,11,14,16)(18,26)(24,29)(25,42)(27,28)(33,43)(39,45)(40,44)(41,46)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=i,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=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185712);