#263347

order	512
nilpotency class	3
isoclinism	isoclinic class of #263331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

826 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=di,f^c=fi

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, 170543);