#278634

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

496 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eh,c^2=i,d^2=ehi,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, 185830);