#343102

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 671 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) = 256

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

Polycyclic presentation:

2^9|a^2=efh,c^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, 250298);