#104890

order	512
nilpotency class	3
isoclinism	isoclinic class of #104877
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

407 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,3)(5,7,6,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(20,21)(24,31)(25,29)(27,28);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,24,28,31)(20,29,21,23)(26,30,27,32);(2,7)(3,8)(5,6)(13,16)(17,19,21,26,22,28,20,27)(18,24,29,32,25,31,23,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fgh,e^c=egh,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12086);