#101047

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

646 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,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, 8243);