#101891

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

331 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9087);