#106067

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

982 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13263);