#110002

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17198);