#239016

order	512
nilpotency class	3
isoclinism	isoclinic class of #239009
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

805 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,4)(5,9)(6,10)(7,11)(8,15)(12,14)(13,16)(17,18,23,28)(19,29,33,40)(20,34,37,25)(21,27,22,26)(24,31,41,32)(30,45,47,42)(35,48,46,39)(36,43,38,44);(1,3)(2,4)(5,12)(6,8)(7,13)(9,14)(10,15)(11,16)(17,19,23,33)(18,24,28,41)(20,35,37,46)(21,31,22,32)(25,42,34,45)(26,40,27,29)(30,36,47,38)(39,43,48,44);(1,4)(2,8)(3,9)(5,14)(6,15)(7,16)(10,13)(11,12)(17,20,21,36,23,37,22,38)(18,25,26,43,28,34,27,44)(19,30,32,46,33,47,31,35)(24,39,29,45,41,48,40,42);(1,5,7,6)(2,9,11,10)(3,12,13,8)(4,14,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,43,34,44)(30,35,47,46)(39,42,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,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, 146212);