#333143

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18686	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240339);