#345893

order	512
nilpotency class	3
isoclinism	isoclinic class of #345888
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

497 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 114 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)(4,8)(5,7)(9,13)(10,14)(11,15)(12,16)(17,18)(19,28)(20,33)(21,24)(22,40)(23,27)(25,30)(26,37)(29,41)(31,39)(32,45)(34,42)(35,36)(38,43)(44,46)(47,48);(3,9)(4,11)(6,13)(8,15)(10,12)(14,16)(17,19,22,30,24,32,23,31)(18,25,27,28,21,39,40,45)(20,34,37,29,38,46,36,47)(26,42,33,48,35,44,43,41);(1,3,5,10)(2,6,7,14)(4,12,11,9)(8,16,15,13)(17,20)(18,26)(19,29)(21,35)(22,36)(23,37)(24,38)(25,41)(27,43)(28,44)(30,34)(31,46)(32,47)(33,40)(39,48)(42,45);(2,7)(6,14)(8,15)(13,16)(17,21,24,18)(19,28,32,45)(20,35,38,26)(22,27,23,40)(25,31,39,30)(29,44,47,42)(33,36,43,37)(34,41,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253089);