#331074

order	512
nilpotency class	3
isoclinism	isoclinic class of #331055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238270);