#263348

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	#62359	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 892 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)(5,8)(11,12)(15,16)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,32)(28,41)(30,39)(34,45)(36,42)(37,44)(38,46)(40,43)(47,48);(1,2,8,13,6,4,5,7)(3,9,10,16,12,14,11,15)(17,19,21,29,22,30,32,42)(18,23,25,27,26,36,35,39)(20,28,33,43,34,44,46,48)(24,37,31,40,38,41,45,47);(1,3,6,12)(2,9,4,14)(5,11,8,10)(7,15,13,16)(23,36)(27,39)(37,41)(40,47);(1,4,8,7,6,2,5,13)(3,9,10,16,12,14,11,15)(17,20,32,46,22,34,21,33)(18,24,35,45,26,38,25,31)(19,28,42,48,30,44,29,43)(23,37,39,47,36,41,27,40)

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=eh,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, 170544);