#111341

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

878 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

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

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

Polycyclic presentation:

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

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, 18537);