#254212

order	512
nilpotency class	3
isoclinism	isoclinic class of #254179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 161408);