#327938

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
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

654 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)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,31,29,32)(33,34,39,43)(35,40,36,45)(37,42,38,47)(41,44,48,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,30)(18,23,25,27)(20,29,22,28)(24,31,26,32)(33,35)(34,40)(36,39)(37,44)(38,46)(41,47)(42,48)(43,45);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,24)(19,28)(20,22)(25,26)(29,30)(31,32)(33,36,39,35)(34,41,43,48)(37,46,38,44)(40,47,45,42);(4,12)(7,14)(10,16)(13,15)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,31,27,32)(33,37,39,38)(34,42,43,47)(35,44,36,46)(40,48,45,41)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,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, 235134);