#238840

order	512
nilpotency class	3
isoclinism	isoclinic class of #238834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

822 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 182 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,27,30,42)(20,31,33,24)(21,35,37,25)(23,36,39,29)(28,44,32,43)(34,41,48,46)(38,45,40,47);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19,22,30)(18,23,26,39)(20,32,33,28)(21,36,37,29)(24,40,31,38)(25,42,35,27)(34,45,48,47)(41,43,46,44);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,34,22,33,37,48)(18,24,25,41,26,31,35,46)(19,28,29,45,30,32,36,47)(23,38,27,43,39,40,42,44);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(20,33)(24,31)(28,32)(34,48)(38,40)(41,46)(43,44)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146036);