#332030

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3549	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

537 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239226);