#254193

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	#2535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 797 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,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, 161389);