#327950

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,22)(20,28)(21,27)(23,25)(24,30)(26,32)(29,31)(33,34)(35,40)(36,46)(37,43)(38,42)(39,44)(41,47)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,18)(20,29,31,28)(21,27,23,25)(24,32,26,30)(33,35)(34,40)(36,39)(37,45)(38,47)(41,42)(43,48)(44,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,22,31)(18,24,19,26)(21,32,23,30)(25,29,27,28)(33,36,39,35)(34,41,44,48)(37,47,38,45)(40,42,46,43);(4,12)(7,14)(10,16)(13,15)(17,21,22,23)(18,25,19,27)(20,30,31,32)(24,28,26,29)(33,37,39,38)(34,42,44,43)(35,45,36,47)(40,41,46,48)

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=de,f^c=fh,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, 235146);