#343101

order	512
nilpotency class	3
isoclinism	isoclinic class of #343099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18686	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

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, 250297);