#214057

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

745 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121253);