#345867

order	512
nilpotency class	3
isoclinism	isoclinic class of #345848
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	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

494 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,9)(4,7)(5,8)(6,11)(10,16)(12,15)(13,14)(17,18,24,21)(19,29,34,31)(20,35,40,44)(22,28,23,27)(25,33,41,32)(26,39,37,38)(30,45,48,42)(36,46,47,43);(3,10)(5,12)(6,14)(8,15)(9,16)(11,13)(17,19,23,33,24,34,22,32)(18,25,28,31,21,41,27,29)(20,36,38,30,40,47,39,48)(26,43,35,45,37,46,44,42);(1,3,4,11)(2,6,7,9)(5,13,12,10)(8,16,15,14)(17,20)(18,26)(19,30)(21,37)(22,38)(23,39)(24,40)(25,42)(27,35)(28,44)(29,43)(31,46)(32,47)(33,36)(34,48)(41,45);(2,7)(6,9)(8,15)(14,16)(17,21,24,18)(19,31,34,29)(20,37,40,26)(22,27,23,28)(25,32,41,33)(30,46,48,43)(35,39,44,38)(36,42,47,45)

Polycyclic presentation:

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

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, 253063);