#232699

order	512
nilpotency class	3
isoclinism	isoclinic class of #232688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1218 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18,23,28)(19,29,33,39)(20,25,36,44)(21,26,22,27)(24,32,40,31)(30,45,46,41)(34,38,47,48)(35,42,37,43);(7,15)(10,16)(17,19,23,33)(18,24,28,40)(20,34,36,47)(21,32,22,31)(25,41,44,45)(26,29,27,39)(30,37,46,35)(38,43,48,42);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,20,23,36)(18,25,28,44)(19,30,33,46)(21,35,22,37)(24,38,40,48)(26,42,27,43)(29,45,39,41)(31,34,32,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,44,43)(30,34,46,47)(38,41,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139895);