#345815

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
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

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

normal subgroups only

This group has 504 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)(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);(3,10)(5,12)(6,14)(8,15)(9,16)(11,13)(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);(1,3,4,11)(2,6,7,9)(5,13,12,10)(8,16,15,14)(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)(6,9)(8,15)(14,16)(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=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=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

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, 253011);