#190021

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

885 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 276 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,g^b=gh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97217);