#278627

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
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	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

495 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,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=ghi

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, 185823);