#238941

order	512
nilpotency class	3
isoclinism	isoclinic class of #238935
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 826 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:

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

Polycyclic presentation:

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

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, 146137);