#111342

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

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	C₂×#3147	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1210 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 202 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)(5,6,8,10,9,12,7,11)(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)(6,10)(7,8)(11,12)(13,15)(14,19)(16,28)(17,31)(18,26)(20,36)(21,23)(22,39)(24,29)(25,33)(27,44)(30,43)(32,42)(34,37)(35,38)(40,41);(6,10)(7,8)(11,12)(13,16)(14,20)(15,24)(17,32)(18,30)(19,34)(21,27)(22,40)(23,41)(25,43)(26,42)(28,31)(29,33)(35,44)(36,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=h,g^2=i,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, 18538);