#328154

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

700 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)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)(33,34,37,41)(35,42,44,39)(36,40,38,46)(43,47,45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(33,35)(34,39)(36,45)(37,44)(38,43)(40,47)(41,42)(46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(35,43)(36,38)(39,47)(40,46)(42,48)(44,45);(4,12)(7,14)(10,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(33,36,37,38)(34,40,41,46)(35,43,44,45)(39,48,42,47)

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235350);