#289401

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196597);