#333268

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 589 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240464);