#343050

order	512
nilpotency class	3
isoclinism	isoclinic class of #343027
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	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 250246);