#265292

order	512
nilpotency class	3
isoclinism	isoclinic class of #265283
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	C₂³×C₄	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

859 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,24,30)(19,31,21,39)(20,35,38,44)(22,29,23,28)(25,34,27,33)(26,40,43,37)(32,42,36,41)(45,47,48,46);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,23,34,24,21,22,33)(18,25,29,39,30,27,28,31)(20,32,40,47,38,36,37,46)(26,41,44,48,43,42,35,45);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20)(18,26)(19,32)(21,36)(22,37)(23,40)(24,38)(25,41)(27,42)(28,35)(29,44)(30,43)(31,45)(33,46)(34,47)(39,48);(17,21,23,33,24,19,22,34)(18,27,29,31,30,25,28,39)(20,36,40,46,38,32,37,47)(26,42,44,45,43,41,35,48)

Polycyclic presentation:

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

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, 172488);