#101329

order	512
nilpotency class	3
isoclinism	isoclinic class of #101243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

370 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=g,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eg,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525);