#323217

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=h,d^2=e,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=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230413);